The Robin Problems in the Coupled System of Wave Equations on a Half-Line

Abstract

This article investigates the local well-posedness of a coupled system of wave equations on a half-line, with a particular emphasis on Robin boundary conditions within Sobolev spaces. We provide estimates for the solutions to linear initial-boundary-value problems related to the coupled system of wave equations, utilizing the Unified Transform Method in conjunction with the Hadamard norm while considering the influence of external forces. Furthermore, we demonstrate that replacing the external force with a nonlinear term alters the iteration map defined by the unified transform solutions, making it a contraction map in a suitable solution space. By employing the contraction mapping theorem, we establish the existence of a unique solution. Finally, we show that the data-to-solution map is locally Lipschitz continuous, thus confirming the local well-posedness of the coupled system of wave equations under consideration.

1. Introduction and Main Results

1.1. Introduction

In this article, we consider the following system of coupled wave equations with Robin boundary conditions:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{u}_{tt}=u_{xx}+u^{2}v,\hfill & x\in (0,\infty ),t\in (0,T),\hfill \\ v_{tt}=v_{xx}+u{v}^2,\hfill & x\in (0,\infty ),t\in (0,T),\hfill \\ (u,v)(x,0)=(u_0,v_0)\left(x\right),\hfill & x\in [0,\infty ),\hfill \\ (u_t,v_t)(x,0)=(u_{*},v_{*})\left(x\right),\hfill & x\in [0,\infty ),\hfill \\ u_x(0,t)-{\varpi }_{1}u(0,t)={\gamma }_1\left(t\right),\hfill & t\in [0,T],\hfill \\ v_x(0,t)-{\varpi }_{2}v(0,t)={\gamma }_2\left(t\right),\hfill & t\in [0,T],\hfill \end{array}\right.\end{array}$$

(1)

where $0<T<1$ and ${\varpi }_i\ge 1$ for $i=1,2$; $u(x,t)$ and $v(x,t)$ are real-valued functions; $u_0\left(x\right)$, $v_0\left(x\right)$, $u_{*}\left(x\right)$, $v_{*}\left(x\right)$ are initial data in $H_x^s(0,\infty )$; and ${\gamma }_1\left(t\right),{\gamma }_2\left(t\right)$ are boundary data in $H_t^{s-1}(0,T)$. The Robin boundary conditions model the dynamic balance of partial energy reflection and absorption at a system’s boundary. It is widely applied in simulating interactions in fields such as acoustics, electromagnetics, elastic waves, and heat conduction.

The primary objective of this article is to employ the Unified Transform Method (UTM) to derive UTM formulas for the linear initial-boundary-value problems (IBVPs) associated with the system (1). We then provide estimates for these UTM formulas and utilize them to construct an iteration map in a suitable function space, which allows us to prove the local well-posedness of (1). For readers interested in UTM and its application to proving the local well-posedness of various equations, please refer to the articles [1,2,3,4,5,6].

Typically, the solution of IBVPs for linear partial differential equations with constant coefficients involves methods such as the separation of variables and specific integral transforms. These classical methods apply only to problems with particular boundary conditions. Fokas’ UTM not only handles the problems solvable by these classical methods but also extends to address issues that classical methods cannot resolve. Moreover, the UTM can clearly determine which boundary value problems are well-posed. Additionally, the paper [7] highlights several key benefits of the UTM compared to standard methods:

• Efficiency: The UTM offers a more efficient means of providing explicit solutions compared to standard methods, and it is capable of addressing problems involving higher-order derivatives that classical approaches may not handle effectively.
• Unified Approach: The UTM provides a consistent framework applicable to a wide range of problems, facilitating the determination of necessary boundary conditions for well-posedness, even in complex scenarios.
• Flexible Evaluation: Solutions obtained through the UTM can be efficiently evaluated using various techniques, including integration path parameterization, asymptotic methods, or the residue theorem.
• Minimal Knowledge Required: The UTM necessitates only a basic understanding of Fourier transforms, the residue theorem, and Jordan’s lemma, making it accessible to those with fundamental knowledge in these areas.

Based on the advantages of the UTM outlined above, this article applies the UTM to derive the UTM formulas for the linear wave equations on the half-line. We then investigate whether these formulas remain valid for data in broader, appropriate Sobolev spaces and conduct a deeper analysis of the linear wave equations on the half-line. This approach ultimately helps us prove the local well-posedness of (1).

Next, we explain the motivation behind considering the system (1). From a mathematical perspective, we are curious whether the UTM can help us establish the local well-posedness of the following system:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{u}_{tt}=u_{xx}+f(u,v),\hfill \\ v_{tt}=v_{xx}+g(u,v),\hfill \end{array}\right.\end{array}$$

(2)

where f and g are polynomial functions of u and v. Furthermore, we can also consider f and g as continuous functions of u and v.

Given that many results already exist on the local well-posedness of the coupled Klein–Gordon equations, we chose to study the distinct system (1). Our subsequent research will investigate the system (2).

The coupled system of Klein–Gordon equations is

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{u}_{tt}=u_{xx}+h_u(u,v),\hfill \\ v_{tt}=d^2{v}_{xx}+h_v(u,v),\hfill \end{array}\right.\end{array}$$

where $h(u,v)$ is a nonlinear coupling potential function, and d is the ratio of the sound speeds in the u- and v-components. The system describes the long-wave dynamics of a pair of coupled one-dimensional periodic particle chains [8]. The elements of each chain are connected through linear coupling, while the chains interact via nonlinear coupling.

When $h(u,v)={\displaystyle \frac{a^2{u}^2}{2}}+{\displaystyle \frac{b^2{v}^2}{2}}+{\displaystyle \frac{c^2{u}^2{v}^2}{2}}$, where u and v are scalar fields of masses a and b, respectively, and c is the interaction constant, the system models the motion of charged mesons in an electromagnetic field. For readers interested in the coupled system of Klein–Gordon equations, please refer to [9,10,11,12,13,14].

We now present recent studies that explore the existence, uniqueness, and well-posedness of solutions for coupled systems of wave equations. In [15], the authors establish the well-posedness and exponential stability of a strongly coupled Klein–Gordon system in a bounded domain with smooth boundaries, considering the effect of locally distributed viscoelastic damping. In their research, the authors employ microlocal analysis tools and a unique continuation principle to address the challenges posed by integral terms. They also formulate a perturbation problem to obtain the exponential decay rate. The conclusions show that under suitable initial conditions, the system achieves exponential energy decay, thereby proving the stability of the system.

In [16], the authors investigated the local well-posedness of the one-dimensional nonlinear wave equation on a half-line. They used the UTM to help analyze the problem’s local well-posedness.

In [17], the authors study a coupled system of two biharmonic equations with damping and source terms of variable-exponent nonlinearities under mixed boundary value conditions. They investigate the existence and uniqueness of weak solutions. By applying Green’s formula, the authors derive an integral equation for the weak solution and then prove the existence and uniqueness of the solution using the fixed-point theorem. Additionally, they establish a blow-up result for negative-initial energy solutions within a finite time.

In [18], the authors investigate the global existence, uniqueness, and uniform stability of the solution energy for the Klein–Gordon system. They employ the Faedo–Galerkin method with compactness arguments and the energy method to prove the existence and uniqueness of the global solution. Additionally, they demonstrate that under specific boundary conditions, the solutions of the system exhibit stable asymptotic behavior over time.

In [19], the authors study the global existence theory for a system of wave equations with variable-exponent nonlinearities. They use the Galerkin method and compactness properties to prove the existence of solutions and further demonstrate that these solutions stabilize towards a stationary state under sufficient regularity conditions.

In [20], the authors investigate the existence of periodic solutions for a Klein–Gordon system with linear coupling and prove that as the coupling constant approaches zero, these solutions converge to those of the uncoupled wave equation. The results highlight the high regularity and stability of the solutions.

In [21], the authors investigate a coupled system of wave equations with coupled Robin boundary conditions, employing group methods to study the issues of approximate boundary controllability and approximate boundary synchronization. The authors provide an algebraic lemma, which generalizes Kalman’s criterion or the Hautus test. They also establish the well-posedness of the coupled system of wave equations with coupled Robin boundary conditions. Additionally, the authors demonstrate the equivalence between the approximate controllability of the system and the D-observability of its adjoint system under certain conditions.

In [22], the authors study a coupled system of two nonlinear hyperbolic equations, where the exponents in the damping and source terms are variables. They establish the existence and uniqueness of solutions by using the Faedo–Galerkin approximation and the Banach fixed-point theorem to prove the existence and uniqueness of weak solutions. Additionally, they apply the energy method to demonstrate that certain solutions with positive-initial energy blow-up in a finite time.

In [23], the authors analyze the solutions of the nonlinear Klein–Gordon equation, presenting necessary and sufficient conditions for finite-time blow-up based on new mass properties of ordinary differential equations. They refine Levine’s concavity method and also prove the global existence of solutions and their asymptotic behavior under specific initial energy conditions.

After introducing the coupled systems of wave equations under consideration, the next subsection will present the main results of this article.

1.2. Main Results

Now, we briefly outline the process through which we establish the local well-posedness of the system (1). The proof is carried out in four steps. First, we replace the nonlinear terms $u^{2}v$ and $v^{2}u$ with external forces, allowing us to derive UTM formulas for the corresponding linear IBVPs. Second, using these UTM formulas, we derive linear estimates that account for the data and forcing terms in appropriate functional spaces. Third, we define an iterative mapping in the chosen solution space, substituting the nonlinear terms for the external forces. We then prove that this mapping is a contraction onto a closed ball $B(0,r)$, ensuring the existence of a unique solution by applying the contraction mapping theorem. Finally, we show that the data-to-solution map is locally Lipschitz continuous, thus confirming the local well-posedness of the IBVP (1).

Before introducing the main theorem of this article, we will briefly define some terms. For $s\in \mathbb{R}$, the Sobolev space $H^s\left(\mathbb{R}\right)$ consists of all tempered distributions F that have a finite norm given by

$$\begin{array}{c}\hfill {\Vert F\Vert }_{H^s\left(\mathbb{R}\right)}\doteq {\left({\int }_{\xi \in \mathbb{R}}{(1+{\xi }^2)}^s{\left|\widehat{F}\left(\xi \right)\right|}^{2}d\xi \right)}^{{\displaystyle \frac{1}{2}}},\end{array}$$

(3)

where $\widehat{F}\left(\xi \right)$ denotes the Fourier transform of F, defined by

$$\begin{array}{c}\hfill \widehat{F}\left(\xi \right)\doteq {\int }_{\mathbb{R}}{e}^{-ix\xi }F\left(x\right)dx.\end{array}$$

Additionally, for any open interval $(a,b)$ in $\mathbb{R}$, the Sobolev space $H^s(a,b)$ is defined as

$$\begin{array}{c}\hfill H^s(a,b)=\left\{\begin{array}{c}{f:f=F|}_{(a,b)},\mathrm{where}F\in H^s\left(\mathbb{R}\right)\hfill \\ {\mathrm{and}\Vert f\Vert }_{H^s(a,b)}\doteq \underset{F\in H^s\left(\mathbb{R}\right)}{inf}{\Vert F\Vert }_{H^s\left(\mathbb{R}\right)}<\infty \hfill \end{array}\right\}.\end{array}$$

When we apply the UTM to solve the forced linear Robin IBVP, it yields the following Fourier transform.

Definition 1

(Half-Line Fourier Transform). Let $\Phi \left(x\right)$ be a test function defined on the interval $(0,\infty )$. The Fourier transform of $\Phi \left(x\right)$, restricted to the half-line, is expressed as

$$\begin{array}{c}\hfill \widehat{\Phi }\left(k\right)\doteq {\int }_0^{\infty }\Phi \left(x\right)e^{-ikx}dx,\end{array}$$

(4)

where $k\in \mathbb{C}$ and $\Im \left(k\right)\le 0$. Here, $\Re \left(k\right)$ and $\Im \left(k\right)$ denote the real and imaginary components of k, respectively.

Remark 1.

In the case of Equation (4), it is straightforward to observe that if Φ is integrable over the interval $(0,\infty )$, the Fourier transform $\widehat{\Phi }\left(k\right)$ is well-defined for values of k where $\Im \left(k\right)\le 0$. Furthermore, the half-line Fourier transform extends naturally within the space $L^2(0,\infty )$. A function $\Phi \in L^2(0,\infty )$ can be extended to the entire real line by defining it as zero for all $x<0$, thereby embedding Φ into $L^2\left(\mathbb{R}\right)$. This allows the half-line Fourier transform to be written similarly to the standard Fourier transform for functions defined on $\mathbb{R}$. Consequently, the inverse transform can also be defined analogously to the inverse Fourier transform on the full real line.

First, we will outline the first step in our strategy for tackling the Robin boundary problem associated with the forced linear wave equations. For the nonlinear IBVP (1), the corresponding linear problem is the forced linear IBVP:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{u}_{tt}=u_{xx}+f(x,t),\hfill & x\in (0,\infty ),t\in (0,T),\hfill \\ v_{tt}=v_{xx}+g(x,t),\hfill & x\in (0,\infty ),t\in (0,T),\hfill \\ (u,v)(x,0)=(u_0,v_0)\left(x\right),\hfill & x\in [0,\infty ),\hfill \\ (u_t,v_t)(x,0)=(u_{*},v_{*})\left(x\right),\hfill & x\in [0,\infty ),\hfill \\ u_x(0,t)-{\varpi }_{1}u(0,t)={\gamma }_1\left(t\right),\hfill & t\in [0,T],\hfill \\ v_x(0,t)-{\varpi }_{2}v(0,t)={\gamma }_2\left(t\right),\hfill & t\in [0,T],\hfill \end{array}\right.\end{array}$$

(5)

where ${\varpi }_i\ge 1$, for $i=1,2$, and $0<T<1$.

The IBVP (5) can be separated into two parts, (6) and (7):

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{u}_t=u_1,\hfill \\ {\left(u_1\right)}_t=u_{xx}+f(x,t),\hfill \\ u(x,0)=u_0\left(x\right),u_1(x,0)=u_{*}\left(x\right),\hfill \\ u_x(0,t)-{\varpi }_{1}u(0,t)={\gamma }_1\left(t\right),\hfill \end{array}\right.\end{array}$$

(6)

and

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{v}_t=v_1,\hfill \\ {\left(v_1\right)}_t=v_{xx}+g(x,t),\hfill \\ v(x,0)=v_0\left(x\right),v_1(x,0)=v_{*}\left(x\right),\hfill \\ v_x(0,t)-{\varpi }_{2}v(0,t)={\gamma }_2\left(t\right).\hfill \end{array}\right.\end{array}$$

(7)

Next, we briefly explain how to use the UTM to derive the corresponding formulas for (6). A similar process can be applied to (7) to obtain its UTM formulas. For Equation (6), we can express it as

$$\begin{array}{c}\hfill {\partial }_{t}Q+\Lambda (-i{\partial }_x)Q=F,\mathrm{where}Q=\left[\begin{array}{c}u\\ u_1\end{array}\right],\Lambda \left(k\right)=\left[\begin{array}{cc}0& -1\\ k^2& 0\end{array}\right],\mathrm{and}F=\left[\begin{array}{c}0\\ f(x,t)\end{array}\right].\end{array}$$

The first step in applying UTM to (6) is to rewrite the above system of equations in a divergence form as

$$\begin{array}{c}\hfill {\left(e^{-ikxI+\Lambda \left(k\right)t}Q\right)}_t-{\left(e^{-ikxI+\Lambda \left(k\right)t}M(x,t,k)Q\right)}_x=e^{-ikxI+\Lambda \left(k\right)t}F,\end{array}$$

(8)

where I is a 2-dimensional identity matrix and the vector $M(x,t,k)$ is a differential matrix operator which is defined by

$$\begin{array}{c}\hfill M(x,t,k)=i{\left.{\displaystyle \frac{\Lambda \left(k\right)-\Lambda \left(l\right)}{k-l}}\right|}_{l=-i{\partial }_x}.\end{array}$$

We refer to this form (8) as the local relation.

In the next step, we integrate the local relation over an infinite strip in the $(x,t)$-plane. Applying Green’s Theorem allows us to convert the area integrals into boundary integrals along the domain’s boundary. Through this process, we obtain the following global relation:

$$\begin{array}{c}\hfill \widehat{Q_0}\left(k\right)-e^{\Lambda \left(k\right)t}\widehat{Q}(k,t)-G(k,t)={\int }_0^t{e}^{\Lambda \left(k\right)t^{\prime }}\widehat{F}(k,t^{\prime })d{t}^{\prime },\end{array}$$

where

$$\begin{array}{cc}\hfill Q_0& =\left[\begin{array}{c}{u}_0\left(x\right)\\ u_{*}\left(x\right)\end{array}\right],\widehat{Q_0}\left(k\right)={\int }_0^{\infty }{e}^{-ikx}{Q}_0\left(x\right)dx,\widehat{Q}\left(k,t\right)={\int }_0^{\infty }{e}^{-ikx}Q(x,t)dx,\hfill \\ \hfill G\left(k,t\right)& ={\int }_0^t{e}^{\Lambda \left(k\right)x}M(0,x,k)Q_0\left(x\right)dx,\widehat{F}\left(k,t\right)={\int }_0^{\infty }{e}^{-ikx}F(x,t)dx.\hfill \end{array}$$

Then, we apply the inverse Fourier transform to the global relation, and we have a solution formula with unknown boundary conditions. We can use the symmetry relation of $\Lambda \left(k\right)$ replacing k by $-k$ to solve the unknown boundary condition; then, we obtain the UTM formulas (9) and (10) for (6). A similar process can be applied to (7) to obtain its UTM formulas (11) and (12).

Due to the complexity and length of the detailed calculations in this section, interested readers are encouraged to refer to references [16,24] for further details.

The UTM formulas for (6) and (7):

$$\begin{array}{cc}\hfill u(x,t)& \doteq S_1[u_0,u_{*},{\gamma }_1;f]\left(x,t\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{\infty }{\displaystyle \frac{i{e}^{ikx}}{2i\sqrt{k^2}}}\left(\sqrt{k^2}{e}^{-i\sqrt{k^2}t}+\sqrt{k^2}{e}^{i\sqrt{k^2}t}\right)\left(\widehat{u_0}\left(k\right)+\left({\displaystyle \frac{{\varpi }_1+ik}{-{\varpi }_1+ik}}\right)\widehat{u_0}\left(-k\right)\right)dk\hfill \\ \hfill & +{\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{\infty }{\displaystyle \frac{e^{ikx}}{2i\sqrt{k^2}}}\left(e^{i\sqrt{k^2}t}-e^{-i\sqrt{k^2}t}\right)\left(\widehat{u_{*}}\left(k\right)+\left({\displaystyle \frac{{\varpi }_1+ik}{-{\varpi }_1+ik}}\right)\widehat{u_{*}}\left(-k\right)\right)dk\hfill \\ \hfill & +{\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{\infty }{\displaystyle \frac{e^{ikx}}{\sqrt{k^2}}}\left({\displaystyle \frac{k}{-{\varpi }_1+ik}}\right)\left(e^{-i\sqrt{k^2}t}\widetilde{{\gamma }_{11}}\left(k,t\right)-e^{i\sqrt{k^2}t}\widetilde{{\gamma }_{12}}\left(k,t\right)\right)dk\hfill \\ \hfill & +{\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{\infty }{\displaystyle \frac{e^{ikx}}{2i\sqrt{k^2}}}\left[e^{i\sqrt{k^2}t}\left({\widehat{f_2}}^{*}\left(k,t\right)+\left({\displaystyle \frac{{\varpi }_1+ik}{-{\varpi }_1+ik}}\right){\widehat{f_2}}^{*}\left(-k,t\right)\right)\right.\hfill \\ \hfill & \left.-e^{-i\sqrt{k^2}t}\left({\widehat{f_1}}^{*}\left(k,t\right)+\left({\displaystyle \frac{{\varpi }_1+ik}{-{\varpi }_1+ik}}\right){\widehat{f_1}}^{*}\left(-k,t\right)\right)\right]dk\hfill \end{array}$$

(9)

and

$$\begin{array}{cc}\hfill u_1\left(x,t\right)& \doteq S_2[u_0,u_{*},{\gamma }_1;f]\left(x,t\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{\infty }{\displaystyle \frac{e^{ikx}}{2i\sqrt{k^2}}}\left(k^2{e}^{-i\sqrt{k^2}t}-k^2{e}^{i\sqrt{k^2}t}\right)\left(\widehat{u_0}\left(k\right)+\left({\displaystyle \frac{{\varpi }_1+ik}{-{\varpi }_1+ik}}\right)\widehat{u_0}\left(-k\right)\right)dk\hfill \\ \hfill & +{\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{\infty }{\displaystyle \frac{e^{ikx}}{2}}\left(e^{-i\sqrt{k^2}t}+e^{i\sqrt{k^2}t}\right)\left(\widehat{u_{*}}\left(k\right)+\left({\displaystyle \frac{{\varpi }_1+ik}{-{\varpi }_1+ik}}\right)\widehat{u_{*}}\left(-k\right)\right)dk\hfill \\ \hfill & +{\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{\infty }-i{e}^{ikx}\left({\displaystyle \frac{k}{-{\varpi }_1+ik}}\right)\left(e^{i\sqrt{k^2}t}\widetilde{{\gamma }_{12}}\left(k,t\right)+e^{-i\sqrt{k^2}t}\widetilde{{\gamma }_{11}}\left(k,t\right)\right)dk\hfill \\ \hfill & +{\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{\infty }{\displaystyle \frac{e^{ikx}}{2}}\left[e^{-i\sqrt{k^2}t}\left({\widehat{f_1}}^{*}\left(k,t\right)+\left({\displaystyle \frac{{\varpi }_1+ik}{-{\varpi }_1+ik}}\right){\widehat{f_1}}^{*}\left(-k,t\right)\right)\right.\hfill \\ \hfill & \left.+e^{i\sqrt{k^2}t}\left({\widehat{f_2}}^{*}\left(k,t\right)+\left({\displaystyle \frac{{\varpi }_1+ik}{-{\varpi }_1+ik}}\right){\widehat{f_2}}^{*}\left(-k,t\right)\right)\right]dk,\hfill \end{array}$$

(10)

where

$${\widehat{f_1}}^{*}\left(k,t\right)={\int }_0^t{\int }_0^{\infty }{e}^{-ikx+i\sqrt{k^2}s}f(x,s)dxds,{\widehat{f_2}}^{*}\left(k,t\right)={\int }_0^t{\int }_0^{\infty }{e}^{-ikx-i\sqrt{k^2}s}f(x,s)dxds,$$

$$\widetilde{{\gamma }_{11}}\left(k,t\right)={\int }_0^t{e}^{i\sqrt{k^2}s}{\gamma }_1\left(s\right)ds,\widetilde{{\gamma }_{12}}\left(k,t\right)={\int }_0^t{e}^{-i\sqrt{k^2}s}{\gamma }_1\left(s\right)ds$$

and

$$\begin{array}{cc}\hfill v\left(x,t\right)& \doteq S_1[v_0,v_{*},{\gamma }_2;g]\left(x,t\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{\infty }{\displaystyle \frac{i{e}^{ikx}}{2i\sqrt{k^2}}}\left(\sqrt{k^2}{e}^{-i\sqrt{k^2}t}+\sqrt{k^2}{e}^{i\sqrt{k^2}t}\right)\left(\widehat{v_0}\left(k\right)+\left({\displaystyle \frac{{\varpi }_2+ik}{-{\varpi }_2+ik}}\right)\widehat{v_0}\left(-k\right)\right)dk\hfill \\ \hfill & +{\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{\infty }{\displaystyle \frac{e^{ikx}}{2i\sqrt{k^2}}}\left(e^{i\sqrt{k^2}t}-e^{-i\sqrt{k^2}t}\right)\left(\widehat{v_{*}}\left(k\right)+\left({\displaystyle \frac{{\varpi }_2+ik}{-{\varpi }_2+ik}}\right)\widehat{v_{*}}\left(-k\right)\right)dk\hfill \\ \hfill & +{\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{\infty }{\displaystyle \frac{e^{ikx}}{\sqrt{k^2}}}\left({\displaystyle \frac{k}{-{\varpi }_2+ik}}\right)\left(e^{-i\sqrt{k^2}t}\widetilde{{\gamma }_{21}}\left(k,t\right)-e^{i\sqrt{k^2}t}\widetilde{{\gamma }_{22}}\left(k,t\right)\right)dk\hfill \\ \hfill & +{\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{\infty }{\displaystyle \frac{e^{ikx}}{2i\sqrt{k^2}}}\left[e^{i\sqrt{k^2}t}\left({\widehat{g_2}}^{*}\left(k,t\right)+\left({\displaystyle \frac{{\varpi }_2+ik}{-{\varpi }_2+ik}}\right){\widehat{g_2}}^{*}\left(-k,t\right)\right)\right.\hfill \\ \hfill & \left.-e^{-i\sqrt{k^2}t}\left({\widehat{g_1}}^{*}\left(k,t\right)+\left({\displaystyle \frac{{\varpi }_2+ik}{-{\varpi }_2+ik}}\right){\widehat{g_1}}^{*}\left(-k,t\right)\right)\right]dk\hfill \end{array}$$

(11)

and

$$\begin{array}{cc}\hfill v_1\left(x,t\right)& \doteq S_2[v_0,v_{*},{\gamma }_2;g]\left(x,t\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{\infty }{\displaystyle \frac{e^{ikx}}{2i\sqrt{k^2}}}\left(k^2{e}^{-i\sqrt{k^2}t}-k^2{e}^{i\sqrt{k^2}t}\right)\left(\widehat{v_0}\left(k\right)+\left({\displaystyle \frac{{\varpi }_2+ik}{-{\varpi }_2+ik}}\right)\widehat{v_0}\left(-k\right)\right)dk\hfill \\ \hfill & +{\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{\infty }{\displaystyle \frac{e^{ikx}}{2}}\left(e^{-i\sqrt{k^2}t}+e^{i\sqrt{k^2}t}\right)\left(\widehat{v_{*}}\left(k\right)+\left({\displaystyle \frac{{\varpi }_2+ik}{-{\varpi }_2+ik}}\right)\widehat{v_{*}}\left(-k\right)\right)dk\hfill \\ \hfill & +{\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{\infty }{\displaystyle \frac{e^{ikx}}{2i\sqrt{k^2}}}\left({\displaystyle \frac{2k}{-{\varpi }_2+ik}}\right)\left(\sqrt{k^2}{e}^{i\sqrt{k^2}t}\widetilde{{\gamma }_{22}}\left(k,t\right)+\sqrt{k^2}{e}^{-i\sqrt{k^2}t}\widetilde{{\gamma }_{21}}\left(k,t\right)\right)dk\hfill \\ \hfill & +{\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{\infty }{\displaystyle \frac{e^{ikx}}{2}}\left[e^{-i\sqrt{k^2}t}\left({\widehat{g_1}}^{*}\left(k,t\right)+\left({\displaystyle \frac{{\varpi }_2+ik}{-{\varpi }_2+ik}}\right){\widehat{g_1}}^{*}\left(-k,t\right)\right)\right.\hfill \\ \hfill & \left.+e^{i\sqrt{k^2}t}\left({\widehat{g_2}}^{*}\left(k,t\right)+\left({\displaystyle \frac{{\varpi }_2+ik}{-{\varpi }_2+ik}}\right){\widehat{g_2}}^{*}\left(-k,t\right)\right)\right]dk\hfill \end{array}$$

(12)

where

$${\widehat{g_1}}^{*}\left(k,t\right)={\int }_0^t{\int }_0^{\infty }{e}^{-ikx+i\sqrt{k^2}s}g(x,s)dxds,{\widehat{g_2}}^{*}\left(k,t\right)={\int }_0^t{\int }_0^{\infty }{e}^{-ikx-i\sqrt{k^2}s}g(x,s)dxds,$$

$$\widetilde{{\gamma }_{21}}\left(k,t\right)={\int }_0^t{e}^{i\sqrt{k^2}s}{\gamma }_2\left(s\right)ds,\widetilde{{\gamma }_{22}}\left(k,t\right)={\int }_0^t{e}^{-i\sqrt{k^2}s}{\gamma }_2\left(s\right)ds.$$

Next, we present the second step, which involves estimating the Hadamard norm of the UTM Formulas (9)–(12) in connection with the Sobolev norms of the data as well as a suitable norm for the forcing terms. Specifically, we derive the following linear estimates.

Theorem 1

(Linear estimates for the wave equations with Robin boundary condition). Consider the wave Equations (6) and (7). Suppose ${\displaystyle \frac{1}{2}}<s<1$, $0<T<1$, $u_0\in H_x^s(0,\infty )$, $u_{*}\in H_x^s(0,\infty )$, $v_0\in H_x^s(0,\infty )$, $v_{*}\in H_x^s(0,\infty )$, ${\gamma }_1\in H_t^{s-1}(0,T)$, and ${\gamma }_2\in H_t^{s-1}(0,T)$. Then, the UTM Formulas (9)–(12) define the solutions $(u,u_1)$ and $(v,v_1)$ to the forced-linear wave-Equation IBVPs (6) and (7), and they satisfy the following estimates:

$$\begin{array}{cc}\hfill & \underset{t\in [0,T]}{sup}{\Vert u\left(t\right)\Vert }_{H_x^s(0,\infty )}+\underset{x\in [0,\infty )}{sup}{\Vert u\left(x\right)\Vert }_{H_t^s(0,T)}\hfill \\ \hfill & \le C_s\left(\Vert u_0{\Vert }_{H_x^s(0,\infty )}+\Vert u_{*}{\Vert }_{H_x^s(0,\infty )}+\Vert {\gamma }_1{\Vert }_{H_t^{s-1}(0,T)}+\sqrt{T}\underset{t\in [0,T]}{sup}{\Vert f\left(t\right)\Vert }_{H_x^s(0,\infty )}\right),\hfill \end{array}$$

(13)

and

$$\begin{array}{cc}\hfill & \underset{t\in [0,T]}{sup}{\Vert v\left(t\right)\Vert }_{H_x^s(0,\infty )}+\underset{x\in [0,\infty )}{sup}{\Vert v\left(x\right)\Vert }_{H_t^s(0,T)}\hfill \\ \hfill & \le d_s\left(\Vert v_0{\Vert }_{H_x^s(0,\infty )}+\Vert v_{*}{\Vert }_{H_x^s(0,\infty )}+\Vert {\gamma }_2{\Vert }_{H_t^{s-1}(0,T)}+\sqrt{T}\underset{t\in [0,T]}{sup}{\Vert g\left(t\right)\Vert }_{H_x^s(0,\infty )}\right),\hfill \end{array}$$

(14)

where $C_s>0$ and $d_s>0$ are constants depending on s.

In the subsequent third and fourth steps, our objective is to prove the uniqueness of the solution for (1) and to establish the local Lipschitz continuity of the data-to-solution mapping. To facilitate this, we define two Banach spaces, referred to as X and D, under the conditions $s>1/2$ and $0<T^{*}\le T<1$, as detailed below:

$$\begin{array}{c}\hfill X={(C([0,T^{*}];H_x^s(0,\infty ))\cap C([0,\infty );H_t^s(0,T^{*})))}^2\end{array}$$

with the following norm:

$$\begin{array}{cc}\hfill {\Vert (u,v)\Vert }_X& =\underset{t\in [0,T^{*}]}{sup}{\Vert u\left(t\right)\Vert }_{H_x^s(0,\infty )}+\underset{x\in [0,\infty )}{sup}{\Vert u\left(x\right)\Vert }_{H_t^s(0,T^{*})}\hfill \\ \hfill & +\underset{t\in [0,T^{*}]}{sup}{\Vert v\left(t\right)\Vert }_{H_x^s(0,\infty )}+\underset{x\in [0,\infty )}{sup}{\Vert v\left(x\right)\Vert }_{H_t^s(0,T^{*})}.\hfill \end{array}$$

The data space

$$D=H_x^s(0,\infty )\times H_x^s(0,\infty )\times H_x^s(0,\infty )\times H_x^s(0,\infty )\times H_t^{s-1}(0,T)\times H_t^{s-1}(0,T)$$

with the data norm

$$\begin{array}{cc}\hfill \Vert (u_0,u_{*},v_0,v_{*},{\gamma }_1,{\gamma }_2){\Vert }_D& =\Vert u_0{\Vert }_{H_x^s(0,\infty )}+\Vert u_{*}{\Vert }_{H_x^s(0,\infty )}+{\Vert v_0\Vert }_{H_x^s(0,\infty )}\hfill \\ \hfill & +\Vert v_{*}{\Vert }_{H_x^s(0,\infty )}+\Vert {\gamma }_1{\Vert }_{H_t^{s-1}(0,T)}+{\Vert {\gamma }_2\Vert }_{H_t^{s-1}(0,T)}.\hfill \end{array}$$

Having established the above definitions, we now present the main result of this study.

Theorem 2

(Local well-posedness of the coupled wave system on the half-line). Consider the coupled wave system (1). Suppose $1/2<s<1$, $u_0\in H_x^s(0,\infty )$, $v_0\in H_x^s(0,\infty )$, $u_{*}\in H_x^s(0,\infty )$, $v_{*}\in H_x^s(0,\infty )$, ${\gamma }_1\left(t\right)\in H_t^{s-1}(0,T)$, and ${\gamma }_2\left(t\right)\in H_t^{s-1}(0,T)$. Then, there exists $T^{*}$, $0<T^{*}\le T<1$, with

$$\begin{array}{c}\hfill T^{*}=min\left\{T,{\displaystyle \frac{1}{2304{\left(C_s^{*}\right)}^{10}{\Vert (u_0,u_{*},v_0,v_{*},{\gamma }_1,{\gamma }_2)\Vert }_D^4}}\right\},\end{array}$$

such that the coupled wave system (1) has a unique solution $(u,v)\in X$ which satisfies the size estimate

$$\begin{array}{c}\hfill {\Vert (u,v)\Vert }_X\le 2C_s^{*}{\Vert (u_0,u_{*},v_0,v_{*},{\gamma }_1,{\gamma }_2)\Vert }_D,\end{array}$$

where $C_s^{*}>0$ is a constant depending on s.

Furthermore, the data-to-solution map $\{u_0,u_{*},v_0,v_{*},{\gamma }_1,{\gamma }_2\}⟼(u,v)$ is locally Lipschitz continuous.

Based on the aforementioned theorem, we have demonstrated the local well-posedness of the coupled wave system (1).

In this article, we conclude that the UTM can be employed to derive UTM formulas for the linear IBVPs associated with system (1). We then provide estimates for these UTM formulas and use them to construct an iteration map in a suitable function space, ultimately proving Theorem 2 to establish the local well-posedness of the system (1).

This article is organized as follows: In Section 2, we introduce a variety of tools that will be used in the subsequent sections. Section 3 focuses on transforming the IBVP (6) into one with zero initial data and simplified boundary conditions, which facilitates estimating the corresponding solution for the linear wave equation IBVP. These estimates are essential for the proof provided in Section 4, where we conclude the proof of Theorem 1. In Section 5, we define the iteration map and demonstrate that it functions as a contraction mapping onto a closed ball, employing the contraction mapping theorem to establish the uniqueness of the solution. Furthermore, in Lemma 7, we show that the data-to-solution map is locally Lipschitz continuous. Finally, we conclude with the proof of Theorem 2.

2. Preliminary Results

In this section, we present several tools that will be utilized in the following sections.

Remark 2.

The norm of Sobolev–Slodobeckii spaces on time is

$$\begin{array}{c}\hfill {\Vert u\left(x\right)\Vert }_{H_t^s(0,T)}^2=\left\{\begin{array}{cc}{\sum }_{j=0}^{\lfloor s\rfloor }\Vert {\partial }_t^j{u\left(x\right)\Vert }_{L_t^2(0,T)}^2+{\Vert {\partial }_t^{\lfloor s\rfloor }u\left(x\right)\Vert }_{\beta }^2,\hfill & \mathit{for}s\in {\mathbb{R}}^{+}\setminus {\mathbb{Z}}^{+},\hfill \\ {\sum }_{j=0}^{\lfloor s\rfloor }{\Vert {\partial }_t^{j}u\left(x\right)\Vert }_{L_t^2(0,T)}^2,\hfill & \mathit{for}s\in {\mathbb{Z}}^{+},\hfill \end{array}\right.\end{array}$$

where $0<\beta <1$ and $\lfloor s\rfloor =s-\beta \in {\mathbb{Z}}^{+}\cup \left\{0\right\}$. The fractional norm ${\Vert ·\Vert }_{\beta }$ is defined by

$$\begin{array}{c}\hfill {\Vert u\left(x\right)\Vert }_{\beta }^2\doteq {\int }_0^T{\int }_0^{T-t}{\displaystyle \frac{{\left|u(x,t+\zeta )-u(x,t)\right|}^2}{{\zeta }^{1+2\beta }}}d\zeta dt,\forall \beta \in (0,1).\end{array}$$

Lemma 1

([25] Ch.3). For $m_1<m<m_2$, we have

$$\begin{array}{c}\hfill {\Vert f\Vert }_{H^m}\le {(\Vert f\Vert }_{H^{m_1}}{)}^{{\displaystyle \frac{m_2-m}{m_2-m_1}}}{(\Vert f\Vert }_{H^{m_2}}{)}^{{\displaystyle \frac{m-m_1}{m_2-m_1}}}.\end{array}$$

Lemma 2

([25] Ch.3). If $s>{\displaystyle \frac{n}{2}}$, then $H^s\left({\mathbb{R}}^n\right)$ is an algebra with respect to the product of functions. That is, if f and $g\in H^s\left({\mathbb{R}}^n\right)$, then $fg\in H^s\left({\mathbb{R}}^n\right)$ with

$$\begin{array}{c}\hfill {\Vert fg\Vert }_{H^s\left({\mathbb{R}}^n\right)}\le C_s{\Vert f\Vert }_{H^s\left({\mathbb{R}}^n\right)}{\Vert g\Vert }_{H^s\left({\mathbb{R}}^n\right)},\end{array}$$

for some constant $C_s>0$, which is a constant depending on s.

Lemma 3

([26] CH.8). Suppose that $(X,{\mu }_1)$ and $(Y,{\mu }_2)$ are two $\sigma$—finite measure spaces and $F:X\times Y\to \mathbb{R}$ is measurable. Then, Minkowski’s integral inequality is

$$\begin{array}{c}\hfill {\left({\int }_Y{\left|{\int }_{X}F(x,y)d{\mu }_1\left(x\right)\right|}^{p}d{\mu }_2\left(y\right)\right)}^{{\displaystyle \frac{1}{p}}}\le {\int }_X{\left({\int }_Y{\left|F(x,y)\right|}^{p}d{\mu }_2\left(y\right)\right)}^{{\displaystyle \frac{1}{p}}}d{\mu }_1\left(x\right),\end{array}$$

for $1\le p<\infty$.

Theorem 3

([26] CH.13). If $f\in L^1\left({\mathbb{R}}^n\right)$, then at every point x of the Lebesque set of f,

$$\begin{array}{c}\hfill f\left(x\right)=\underset{t\to 0}{lim}{\displaystyle \frac{1}{{\left(2\pi \right)}^n}}{\int }_{{\mathbb{R}}^n}\widehat{f}\left(y\right)e^{ix·y}{e}^{-{t\left|y\right|}^2}dy.\end{array}$$

(15)

In addition, if $\widehat{f}\in L^1\left({\mathbb{R}}^n\right)$, then at every Lebesque point x of f,

$$\begin{array}{c}\hfill f\left(x\right)={\displaystyle \frac{1}{{\left(2\pi \right)}^n}}{\int }_{{\mathbb{R}}^n}\widehat{f}\left(y\right)e^{ix·y}dy.\end{array}$$

(16)

In particular, (15) and (16) hold a.e. in ${\mathbb{R}}^n$ and at every point of continuity of f.

Theorem 4

([27]). For each $f\in L^2$, a function $\widehat{f}\in L^2$, so that the following properties hold:

(i)

If $f\in L^1\cap L^2$;

(ii)

For every $f\in L^2$, $\Vert \widehat{f}{\Vert }_2={\Vert f\Vert }_2$;

(iii)

The mapping $f⟼\widehat{f}$ is a Hilbert space isomorphism of $L^2$ onto $L^2$;

(iv)

The following symmetric relation exists between f and $\widehat{f}$: If

$$\begin{array}{c}\hfill {\varphi }_A\left(t\right)={\int }_{-A}^{A}f\left(x\right)e^{-ixt}dm\left(t\right)\mathrm{and}{\psi }_A\left(x\right)={\int }_{-A}^A\widehat{f}\left(t\right)e^{ixt}dm\left(t\right),\end{array}$$

then

$$\begin{array}{c}\hfill \Vert {\varphi }_A-\widehat{f}{\Vert }_2\to 0\mathrm{and}{\Vert {\psi }_A-f\Vert }_2\to 0\mathrm{as}A\to 0.\end{array}$$

(17)

Theorem 5

([28]). If Ω satisfies a uniform interior cone condition (that is, there exists a fixed cone $k_{\Omega }$ such that each $x\in \Omega$ is the vertex of a cone $k_{\Omega }\left(x\right)\subset \overline{\Omega }$ and congruent to $k_{\Omega }$), then there is an embedding

$$\begin{array}{c}\hfill W^{k,p}(\Omega )\begin{array}{c}\nearrow L^{{\displaystyle \frac{np}{(n-kp)}}},\mathrm{for}kp<n,\hfill \\ \searrow C_B^m(\Omega ),\mathrm{for}0\le m<k-{\displaystyle \frac{n}{p}},\hfill \end{array}\end{array}$$

where

$$\begin{array}{c}\hfill C_B^m(\Omega )=\{u\in C^m(\Omega )|D^{\alpha }u\in L^{\infty }(\Omega )\mathrm{for}\left|\alpha \right|\le m\}.\end{array}$$

To facilitate our calculations and enhance the clarity of our presentation, we introduce the following notations.

Remark 3.

Given two quantities $\mathbb{A}$ and $\mathbb{B}$ that may depend on one or more variables, we denote $\mathbb{A}\lesssim \mathbb{B}$ when there exists a positive constant c such that $\mathbb{A}\le c\mathbb{B}$. If both inequalities $\mathbb{A}\lesssim \mathbb{B}$ and $\mathbb{B}\lesssim \mathbb{A}$ hold, we express this relationship as $\mathbb{A}\simeq \mathbb{B}$.

3. The Reduced Pure Linear Robin Problem and Sobolev Space Estimates

In this section, we will demonstrate Theorem 6, which provides estimates of the solution to the linear wave equation IBVP, and will help us to prove Theorem 1 in Section 4.

We begin our analysis with the fundamental IBVP for the linear wave equation defined on the half-line. This involves considering the homogeneous IBVP with zero initial data and non-zero boundary conditions.

Moreover, we assume that the boundary data ${\gamma }_{\ell }$ is a test function of time, where ${\gamma }_{\ell }\in H_t^{s-1}\left(\mathbb{R}\right)$ is an extension of ${\gamma }_1\in H_t^{s-1}(0,T)$ such that

$$\begin{array}{c}\hfill \Vert {\gamma }_{\ell }{\Vert }_{H_t^{s-1}\left(\mathbb{R}\right)}\le 2{\Vert {\gamma }_1\Vert }_{H_t^{s-1}(0,T)},\mathrm{for}{\displaystyle \frac{1}{2}}<s<1,\end{array}$$

(18)

and ${\gamma }_{\ell }$ is compactly supported within the interval $[0,2]$.

This particular problem, referred to as the reduced pure IBVP, can be expressed as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{w}_t=w_1,\hfill & x\in (0,\infty ),t\ge 0,\hfill \\ {\left(w_1\right)}_t=w_{xx},\hfill & x\in (0,\infty ),t\ge 0,\hfill \\ w(x,0)=0,w_t(x,0)=0,\hfill & x\in [0,\infty ),\hfill \\ w_x(0,t)-{\varpi }_{1}w(0,t)={\gamma }_{\ell }\left(t\right),\hfill & t\ge 0.\hfill \end{array}\right.\end{array}$$

(19)

By employing the UTM Formulas (9) and (10), we can derive the corresponding UTM formulas for the reduced pure initial-boundary-value problem (IBVP) (19):

$$\begin{array}{cc}\hfill w(x,t)& =S_1[0,0,{\gamma }_{\ell };0](x,t)\hfill \\ \hfill & ={\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{\infty }{e}^{ikx}\left({\displaystyle \frac{k}{\left|k\right|(-{\varpi }_1+ik)}}\right)(e^{-i\left|k\right|t}\widetilde{{\gamma }_{{\ell }_1}}(k,t)-e^{i\left|k\right|t}\widetilde{{\gamma }_{{\ell }_2}}(k,t))dk,\hfill \end{array}$$

(20)

and

$$\begin{array}{cc}\hfill w_1(x,t)& =S_2[0,0,{\gamma }_{\ell };0](x,t)\hfill \\ \hfill & =-{\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{\infty }{e}^{ikx}\left({\displaystyle \frac{ik}{-{\varpi }_1+ik}}\right)(e^{-i\left|k\right|t}\widetilde{{\gamma }_{{\ell }_2}}(k,t)+e^{i\left|k\right|t}\widetilde{{\gamma }_{{\ell }_1}}(k,t))dk,\hfill \end{array}$$

(21)

where

$$\begin{array}{c}\hfill \widetilde{{\gamma }_{{\ell }_1}}(k,t)={\int }_0^t{e}^{i\sqrt{k^2}s}{\gamma }_{\ell }\left(s\right)ds,\widetilde{{\gamma }_{{\ell }_2}}(k,t)={\int }_0^t{e}^{-i\sqrt{k^2}s}{\gamma }_{\ell }\left(s\right)ds.\end{array}$$

In the subsequent result, we evaluate the solutions (20) and (21) within the Hadamard space.

Theorem 6.

(Estimates for the Pure IBVP on the Half-Line) Let $1/2<s<3/2$ and consider the boundary data test function ${\gamma }_{\ell }\in H_t^{s-1}\left(\mathbb{R}\right)$, which is compactly supported within the interval $[0,2]$. The solution to the reduced pure initial-boundary-value problem (IBVP) (19) satisfies the following Hadamard space estimates:

$$\begin{array}{cc}\hfill \underset{t\in [0,2]}{sup}{\Vert S_1[0,0,{\gamma }_{\ell };0]\left(t\right)\Vert }_{H_x^s(0,\infty )}& \le {\left(C_1\right)}_s{\Vert {\gamma }_{\ell }\Vert }_{H_t^{s-1}\left(\mathbb{R}\right)},\hfill \end{array}$$

(22)

$$\begin{array}{cc}\hfill \underset{t\in [0,2]}{sup}{\Vert S_2[0,0,{\gamma }_{\ell };0]\left(t\right)\Vert }_{H_x^{s-1}(0,\infty )}& \le {\left(C_1\right)}_s{\Vert {\gamma }_{\ell }\Vert }_{H_t^{s-1}\left(\mathbb{R}\right)},\hfill \end{array}$$

(23)

and time estimates:

$$\begin{array}{cc}\hfill \underset{x\in [0,\infty )}{sup}{\Vert S_1[0,0,{\gamma }_{\ell };0]\left(x\right)\Vert }_{H_t^s(0,2)}& \le {\left(C_1\right)}_s{\Vert {\gamma }_{\ell }\Vert }_{H_t^{s-1}\left(\mathbb{R}\right)},\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill \underset{x\in [0,\infty )}{sup}{\Vert S_2[0,0,{\gamma }_{\ell };0]\left(x\right)\Vert }_{H_t^{s-1}(0,2)}& \le {\left(C_1\right)}_s{\Vert {\gamma }_{\ell }\Vert }_{H_t^{s-1}\left(\mathbb{R}\right)},\hfill \end{array}$$

(25)

where $s\in \mathbb{R}$ and ${\left(C_1\right)}_s>0$ is a constant that depends on s.

Proof. 

We will first present the proof for the space estimate given in (22). To perform this, we will analyze Equation (20):

$$\begin{array}{cc}\hfill w(x,t)& =S_1[0,0,{\gamma }_{\ell };0](x,t)\hfill \\ \hfill & ={\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{\infty }{e}^{ikx}\left({\displaystyle \frac{k}{\left|k\right|(-{\varpi }_1+ik)}}\right)(e^{-i\left|k\right|t}\widetilde{{\gamma }_{{\ell }_1}}(k,t)-e^{i\left|k\right|t}\widetilde{{\gamma }_{{\ell }_2}}(k,t))dk\hfill \\ \hfill & =\underset{\left(A\right)}{\underbrace{{\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^0{e}^{ikx}\left({\displaystyle \frac{-1}{-{\varpi }_1+ik}}\right)(e^{ikt}\widetilde{{\gamma }_{{\ell }_1}}(k,t)-e^{-ikt}\widetilde{{\gamma }_{{\ell }_2}}(k,t))dk}}\hfill \\ \hfill & +\underset{\left(B\right)}{\underbrace{{\displaystyle \frac{1}{2\pi }}{\int }_0^{\infty }{e}^{ikx}\left({\displaystyle \frac{1}{-{\varpi }_1+ik}}\right)(e^{-ikt}\widetilde{{\gamma }_{{\ell }_1}}(k,t)-e^{ikt}\widetilde{{\gamma }_{{\ell }_2}}(k,t))dk}}.\hfill \end{array}$$

Regarding equations $\left(A\right)$ and $\left(B\right)$, we deduce that the space Fourier transform of W is given by

$$\begin{array}{c}\hfill {\widehat{w}}^x(k,t)=\left\{\begin{array}{cc}{\displaystyle \frac{(-1)(e^{ikt}\widetilde{{\gamma }_{{\ell }_1}}(k,t)-e^{-ikt}\widetilde{{\gamma }_{{\ell }_2}}(k,t))}{-{\varpi }_1+ik}},\hfill & k\in (-\infty ,0],\hfill \\ {\displaystyle \frac{(e^{-ikt}\widetilde{{\gamma }_{{\ell }_1}}(k,t)-e^{ikt}\widetilde{{\gamma }_{{\ell }_2}}(k,t))}{-{\varpi }_1+ik}},\hfill & k\in [0,\infty ).\hfill \end{array}\right.\end{array}$$

Applying (3), we derive the following inequality:

$$\begin{array}{cc}\hfill & \Vert S_1[0,0,{\gamma }_{\ell };0](x,t){\Vert }_{H_x^s(0,\infty )}^2={\int }_{\mathbb{R}}{(1+k^2)}^s{\left|{\widehat{w}}^x(k,t)\right|}^{2}dk\hfill \\ \hfill & ={\int }_{-\infty }^0{(1+k^2)}^s|{\widehat{w}}^x(k,t){|}^{2}dk+{\int }_0^{\infty }{(1+k^2)}^s{\left|{\widehat{w}}^x(k,t)\right|}^{2}dk\hfill \\ \hfill & \lesssim {\int }_{-\infty }^0{(1+k^2)}^{s-1}\left({\left|\widetilde{{\gamma }_{{\ell }_1}}(k,t){|}^2+|\widetilde{{\gamma }_{{\ell }_2}}(k,t)\right|}^2\right)dk+{\int }_0^{\infty }{(1+k^2)}^{s-1}\left({\left|\widetilde{{\gamma }_{{\ell }_1}}(k,t){|}^2+|\widetilde{{\gamma }_{{\ell }_2}}(k,t)\right|}^2\right)dk\hfill \\ \hfill & ={\int }_{-\infty }^0{(1+k^2)}^{s-1}({\left|{\int }_0^t{e}^{-iks}{\gamma }_{\ell }\left(s\right)ds\right|}^2+{\left|{\int }_0^t{e}^{iks}{\gamma }_{\ell }\left(s\right)ds\right|}^2)dk\hfill \\ \hfill & +{\int }_0^{\infty }{(1+k^2)}^{s-1}({\left|{\int }_0^t{e}^{iks}{\gamma }_{\ell }\left(s\right)ds|\right|}^2+{\left|{\int }_0^t{e}^{-iks}{\gamma }_{\ell }\left(s\right)ds\right|}^2)dk\hfill \\ \hfill & ={\int }_{-\infty }^0{(1+k^2)}^{s-1}({\left|{\int }_{\mathbb{R}}{e}^{-iks}{\chi }_{[0,t]}\left(s\right){\gamma }_{\ell }\left(s\right)ds\right|}^2+{\left|{\int }_{\mathbb{R}}{e}^{iks}{\chi }_{[0,t]}\left(s\right){\gamma }_{\ell }\left(s\right)ds\right|}^2)dk\hfill \\ \hfill & +{\int }_0^{\infty }{(1+k^2)}^{s-1}({\left|{\int }_{\mathbb{R}}{e}^{iks}{\chi }_{[0,t]}\left(s\right){\gamma }_{\ell }\left(s\right)ds\right|}^2+{\left|{\int }_{\mathbb{R}}{e}^{-iks}{\chi }_{[0,t]}\left(s\right){\gamma }_{\ell }\left(s\right)ds\right|}^2)dk\hfill \\ \hfill & ={\int }_{-\infty }^0{(1+k^2)}^{s-1}({\left|\widehat{{\chi }_{[0,t]}{\gamma }_{\ell }}\left(k\right)\right|}^2+{\left|\widehat{{\chi }_{[0,t]}{\gamma }_{\ell }}(-k)\right|}^2)dk\hfill \\ \hfill & +{\int }_0^{\infty }{(1+k^2)}^{s-1}\left({\left|\widehat{{\chi }_{[0,t]}{\gamma }_{\ell }}(-k){|}^2+|\widehat{{\chi }_{[0,t]}{\gamma }_{\ell }}\left(k\right)\right|}^2\right)dk\hfill \\ \hfill & ={\int }_{\mathbb{R}}{(1+k^2)}^{s-1}|\widehat{{\chi }_{[0,t]}{\gamma }_{\ell }}\left(k\right){|}^{2}dk+{\int }_{\mathbb{R}}{(1+k^2)}^{s-1}{\left|\widehat{{\chi }_{[0,t]}{\gamma }_{\ell }}(-k)\right|}^{2}dk\hfill \\ \hfill & =\Vert {\chi }_{[0,t]}{\gamma }_{\ell }{\Vert }_{H_t^{s-1}\left(\mathbb{R}\right)}^2+\Vert {\chi }_{[0,t]}{\gamma }_{\ell }{\Vert }_{H_t^{s-1}\left(\mathbb{R}\right)}^2\lesssim {\Vert {\gamma }_{\ell }\Vert }_{H_t^{s-1}\left(\mathbb{R}\right)}^2.\hfill \end{array}$$

Hence, we have Equation (22):

$$\begin{array}{c}\hfill \underset{t\in [0,2]}{sup}\Vert S_1[0,0,{\gamma }_{\ell };0]\left(t\right){\Vert }_{H_x^s(0,\infty )}\le {\left(C_1\right)}_s{\Vert {\gamma }_{\ell }\Vert }_{H_t^{s-1}\left(\mathbb{R}\right)}.\end{array}$$

Next, we begin proving the space estimate (23). We examine Equation (21):

$$\begin{array}{cc}\hfill w_1(x,t)& =S_2[0,0,{\gamma }_{\ell };0](x,t)\hfill \\ \hfill & =-{\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{\infty }{e}^{ikx}\left({\displaystyle \frac{ik}{-{\varpi }_1+ik}}\right)(e^{-i\left|k\right|t}\widetilde{{\gamma }_{{\ell }_2}}(k,t)+e^{i\left|k\right|t}\widetilde{{\gamma }_{{\ell }_1}}(k,t))dk\hfill \\ \hfill & =\underset{\left(C\right)}{\underbrace{-{\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^0{e}^{ikx}\left({\displaystyle \frac{ik}{-{\varpi }_1+ik}}\right)(e^{ikt}\widetilde{{\gamma }_{{\ell }_2}}(k,t)+e^{-ikt}\widetilde{{\gamma }_{{\ell }_1}}(k,t))dk}}\hfill \\ \hfill & \underset{\left(D\right)}{\underbrace{-{\displaystyle \frac{1}{2\pi }}{\int }_0^{\infty }{e}^{ikx}\left({\displaystyle \frac{ik}{-{\varpi }_1+ik}}\right)(e^{-ikt}\widetilde{{\gamma }_{{\ell }_2}}(k,t)+e^{ikt}\widetilde{{\gamma }_{{\ell }_1}}(k,t))}}.\hfill \end{array}$$

Regarding Equations $\left(C\right)$ and $\left(D\right)$, we deduce that the space Fourier transform of $W_1$ is given by

$$\begin{array}{c}\hfill {\widehat{w_1}}^x(k,t)=\left\{\begin{array}{c}{\displaystyle \frac{-ik}{-{\varpi }_1+ik}}(e^{ikt}\widetilde{{\gamma }_{{\ell }_2}}(k,t)+e^{-ikt}\widetilde{{\gamma }_{{\ell }_1}}(k,t)),k\in (-\infty ,0],\hfill \\ {\displaystyle \frac{-ik}{-{\varpi }_1+ik}}(e^{-ikt}\widetilde{{\gamma }_{{\ell }_2}}(k,t)+e^{ikt}\widetilde{{\gamma }_{{\ell }_1}}(k,t)),k\in (0,\infty ).\hfill \end{array}\right.\end{array}$$

Applying (3), we derive the following inequality:

$$\begin{array}{cc}\hfill & \Vert S_2[0,0,{\gamma }_{\ell };0](x,t){\Vert }_{H_x^{s-1}(0,\infty )}^2={\int }_{\mathbb{R}}{(1+k^2)}^{s-1}{\left|{\widehat{w_1}}^x(k,t)\right|}^{2}dk\hfill \\ \hfill & ={\int }_{-\infty }^0{(1+k^2)}^{s-1}|{\widehat{w_1}}^x(k,t){|}^{2}dk+{\int }_0^{\infty }{(1+k^2)}^{s-1}{\left|{\widehat{w_1}}^x(k,t)\right|}^{2}dk\hfill \\ \hfill & ={\int }_{-\infty }^0{(1+k^2)}^{s-1}{\left|{\displaystyle \frac{-ik}{-{\varpi }_1+ik}}(e^{ikt}\widetilde{{\gamma }_{{\ell }_2}}(k,t)+e^{-ikt}\widetilde{{\gamma }_{{\ell }_1}}(k,t))\right|}^{2}dk\hfill \\ \hfill & +{\int }_0^{\infty }{(1+k^2)}^{s-1}{\left|{\displaystyle \frac{-ik}{-{\varpi }_1+ik}}(e^{-ikt}\widetilde{{\gamma }_{{\ell }_2}}(k,t)+e^{ikt}\widetilde{{\gamma }_{{\ell }_1}}(k,t))\right|}^{2}dk\hfill \\ \hfill & \lesssim {\int }_{-\infty }^0{(1+k^2)}^{s-1}\left({\left|\widetilde{{\gamma }_{{\ell }_2}}(k,t){|}^2+|\widetilde{{\gamma }_{{\ell }_1}}(k,t)\right)}^2\right)dk+{\int }_0^{\infty }{(1+k^2)}^{s-1}\left({\left|\widetilde{{\gamma }_{{\ell }_2}}(k,t){|}^2+|\widetilde{{\gamma }_{{\ell }_1}}(k,t)\right)}^2\right)dk\hfill \\ \hfill & ={\int }_{-\infty }^0{(1+k^2)}^{s-1}({\left|{\int }_0^t{e}^{-i\sqrt{k^2}s}{\gamma }_{\ell }\left(s\right)ds\right|}^2+{\left|{\int }_0^t{e}^{i\sqrt{k^2}s}{\gamma }_{\ell }\left(s\right)ds\right|}^2)dk\hfill \\ \hfill & +{\int }_0^{\infty }{(1+k^2)}^{s-1}({\left|{\int }_0^t{e}^{-i\sqrt{k^2}s}{\gamma }_{\ell }\left(s\right)ds\right|}^2+{\left|{\int }_0^t{e}^{i\sqrt{k^2}s}{\gamma }_{\ell }\left(s\right)ds\right|}^2)dk\hfill \\ \hfill & ={\int }_{-\infty }^0{(1+k^2)}^{s-1}({\left|{\int }_{\mathbb{R}}{e}^{iks}{\chi }_{[0,t]}\left(s\right){\gamma }_{\ell }\left(s\right)ds\right|}^2+{\left|{\int }_{\mathbb{R}}{e}^{-iks}{\chi }_{[0,t]}\left(s\right){\gamma }_{\ell }\left(s\right)ds\right|}^2)dk\hfill \\ \hfill & +{\int }_0^{\infty }{(1+k^2)}^{s-1}({\left|{\int }_{\mathbb{R}}{e}^{-iks}{\chi }_{[0,t]}\left(s\right){\gamma }_{\ell }\left(s\right)ds\right|}^2+{\left|{\int }_{\mathbb{R}}{e}^{iks}{\chi }_{[0,t]}\left(s\right){\gamma }_{\ell }\left(s\right)ds\right|}^2)dk\hfill \\ \hfill & ={\int }_{-\infty }^{\infty }{(1+k^2)}^{s-1}|\widehat{{\chi }_{[0,t]}{\gamma }_{\ell }}\left(k\right){|}^{2}dk+{\int }_{-\infty }^{\infty }{(1+k^2)}^{s-1}{\left|\widehat{{\chi }_{[0,t]}{\gamma }_{\ell }}(-k)\right|}^{2}dk\hfill \\ \hfill & =\Vert {\chi }_{[0,t]}{\gamma }_{\ell }{\Vert }_{H_t^{s-1}\left(\mathbb{R}\right)}^2+\Vert {\chi }_{[0,t]}{\gamma }_{\ell }{\Vert }_{H_t^{s-1}\left(\mathbb{R}\right)}^2\lesssim {\Vert {\gamma }_{\ell }\Vert }_{H_t^{s-1}\left(\mathbb{R}\right)}^2.\hfill \end{array}$$

Hence, we have Equation (23):

$$\begin{array}{c}\hfill \underset{[0,2]}{sup}\Vert S_2[0,0,{\gamma }_{\ell };0](x,t){\Vert }_{H_x^{s-1}(0,\infty )}\le {\left(C_1\right)}_s{\Vert {\gamma }_{\ell }\Vert }_{H_t^{s-1}\left(\mathbb{R}\right)}.\end{array}$$

Next, we proceed to prove the time estimates (24) and (25). First, we consider the estimate of $\Vert S_1[0,0,{\gamma }_{\ell };0](x,t){\Vert }_{H_t^s(0,2)}$ as follows:

$$\begin{array}{cc}\hfill & \Vert S_1[0,0,{\gamma }_{\ell };0](x,t){\Vert }_{H_t^s(0,2)}\hfill \\ \hfill & =\Vert {\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^0{e}^{ikx}\left({\displaystyle \frac{-1}{-{\varpi }_1+ik}}\right)(e^{ikt}\widetilde{{\gamma }_{{\ell }_1}}(k,t)-e^{-ikt}\widetilde{{\gamma }_{{\ell }_2}}(k,t))dk\hfill \\ \hfill & +{\displaystyle \frac{1}{2\pi }}{\int }_0^{\infty }{e}^{ikx}\left({\displaystyle \frac{1}{-{\varpi }_1+ik}}\right)(e^{-ikt}\widetilde{{\gamma }_{{\ell }_1}}(k,t)-e^{ikt}\widetilde{{\gamma }_{{\ell }_2}}(k,t)){dk\Vert }_{H_t^s(0,2)},\hfill \\ \hfill & \le \underset{\left(E\right)}{\underbrace{\Vert {\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^0{e}^{ikt}{e}^{ikx}\left({\displaystyle \frac{-1}{-{\varpi }_1+ik}}\right)\widetilde{{\gamma }_{{\ell }_1}}(k,t){dk\Vert }_{H_t^s(0,2)}}}+\underset{\left(F\right)}{\underbrace{\Vert {\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^0{e}^{-ikt}{e}^{ikx}\left({\displaystyle \frac{-1}{-{\varpi }_1+ik}}\right)\widetilde{{\gamma }_{{\ell }_2}}(k,t){dk\Vert }_{H_t^s(0,2)}}}\hfill \\ \hfill & +\underset{\left(G\right)}{\underbrace{\Vert {\displaystyle \frac{1}{2\pi }}{\int }_0^{\infty }{e}^{-ikt}{e}^{ikx}\left({\displaystyle \frac{1}{-{\varpi }_1+ik}}\right)\widetilde{{\gamma }_{{\ell }_1}}(k,t){dk\Vert }_{H_t^s(0,2)}}}+\underset{\left(H\right)}{\underbrace{\Vert {\displaystyle \frac{1}{2\pi }}{\int }_0^{\infty }{e}^{ikt}{e}^{ikx}\left({\displaystyle \frac{1}{-{\varpi }_1+ik}}\right)\widetilde{{\gamma }_{{\ell }_2}}(k,t){dk\Vert }_{H_t^s(0,2)}}}.\hfill \end{array}$$

Applying Theorem 3, we derive the following inequality:

$$\begin{array}{cc}\hfill {\left(E\right)}^2& \le \Vert {\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^0{e}^{ikt}{e}^{ikx}\left({\displaystyle \frac{-1}{-{\varpi }_1+ik}}\right)\widetilde{{\gamma }_{{\ell }_1}}(k,t){dk\Vert }_{H_t^s\left(\mathbb{R}\right)}^2\hfill \\ \hfill & \lesssim {\int }_{\mathbb{R}}{(1+k^2)}^{s-1}|\widehat{{\chi }_{[0,t]}{\gamma }_{\ell }}\left(k\right){|}^{2}dk\le {\Vert {\gamma }_{\ell }\Vert }_{H_t^{s-1}\left(\mathbb{R}\right)}^2,\hfill \\ \hfill {\left(F\right)}^2& \le \Vert {\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^0{e}^{-ikt}{e}^{ikx}\left({\displaystyle \frac{-1}{-{\varpi }_1+ik}}\right)\widetilde{{\gamma }_{{\ell }_2}}(k,t){dk\Vert }_{H_t^s\left(\mathbb{R}\right)}^2\hfill \\ \hfill & =\Vert {\displaystyle \frac{1}{2\pi }}{\int }_0^{\infty }{e}^{ikt}{e}^{-ikx}\left({\displaystyle \frac{-1}{-{\varpi }_1-ik}}\right)\widetilde{{\gamma }_{{\ell }_2}}(k,t){dk\Vert }_{H_t^s\left(\mathbb{R}\right)}^2\hfill \\ \hfill & \lesssim {\int }_{\mathbb{R}}{(1+k^2)}^{s-1}|\widehat{{\chi }_{[0,t]}{\gamma }_{\ell }}\left(k\right){|}^{2}dk\le {\Vert {\gamma }_{\ell }\Vert }_{H_t^{s-1}\left(\mathbb{R}\right)}^2.\hfill \end{array}$$

Following a similar proof process as used for the estimates of ${\left(E\right)}^2$ and ${\left(F\right)}^2$ above, we also derive the following results:

$$\begin{array}{c}\hfill {\left(G\right)}^2\le \Vert {\gamma }_{\ell }{\Vert }_{H_t^{s-1}\left(\mathbb{R}\right)}^2,{\left(H\right)}^2\le {\Vert {\gamma }_{\ell }\Vert }_{H_t^{s-1}\left(\mathbb{R}\right)}^2.\end{array}$$

Thus, we can conclude that

$$\begin{array}{c}\hfill \Vert S_1[0,0,{\gamma }_{\ell };0](x,t){\Vert }_{H_t^s(0,2)}\lesssim {\Vert {\gamma }_{\ell }\Vert }_{H_t^{s-1}\left(\mathbb{R}\right)},\end{array}$$

and then, we have Equation (24):

$$\begin{array}{c}\hfill \underset{x\in [0,\infty )}{sup}\Vert S_1[0,0,{\gamma }_{\ell };0]\left(x\right){\Vert }_{H_t^s(0,2)}\le {\left(C_1\right)}_s{\Vert {\gamma }_{\ell }\Vert }_{H_t^{s-1}\left(\mathbb{R}\right)}.\end{array}$$

Following a similar proof process as in (24), we can derive Equation (25):

$$\begin{array}{c}\hfill \underset{x\in [0,\infty )}{sup}\Vert S_2[0,0,{\gamma }_{\ell };0]\left(x\right){\Vert }_{H_t^{s-1}(0,2)}\le {\left(C_1\right)}_s{\Vert {\gamma }_{\ell }\Vert }_{H_t^{s-1}\left(\mathbb{R}\right)}.\end{array}$$

We finish the proof of Theorem 6. □

4. The Proof of Theorem 1 (about the Forced Linear IBVP Estimates)

In this section, we utilize Theorem 6 to perform the space and time estimates for (36) and (38). We establish the fundamental linear estimates (13) and (14), thereby completing the proof of Theorem 1. We begin by breaking down the forced linear IBVP (6) into a combination of simple IVPs and IBVPs.

4.1. Decomposition into a Superposition of IVPs and IBVPs

In this subsection, we focus on proving Theorem 1. Our approach begins with decomposing the forced linear IBVP (6) into a superposition of the following problems.

Let $U_0\in H_x^s\left(\mathbb{R}\right)$ and $U_{*}\in H_x^s\left(\mathbb{R}\right)$ represent extensions of the initial data $u_0\in H_x^s(0,\infty )$ and $u_{*}\in H_x^s(0,\infty )$, respectively, such that

$$\begin{array}{cc}\hfill \Vert U_0{\Vert }_{H_x^s\left(\mathbb{R}\right)}& \le 2\Vert u_0{\Vert }_{H_x^s(0,\infty )},s\ge 0,\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill \Vert U_{*}{\Vert }_{H_x^s\left(\mathbb{R}\right)}& \le 2\Vert u_{*}{\Vert }_{H_x^s(0,\infty )},s\ge 0,\hfill \end{array}$$

(27)

and F be an extension of the forcing term f such that

$$\begin{array}{c}\hfill \underset{t\in [0,T]}{sup}{\Vert F\left(t\right)\Vert }_{H_x^s\left(\mathbb{R}\right)}\le 2\underset{t\in [0,T]}{sup}{\Vert f\left(t\right)\Vert }_{H_x^s(0,\infty )},s>0.\end{array}$$

(28)

Based on the definitions of $U_0$, $U_{*}$, and F established, we initiate our approach by breaking down the IBVP (6) into a combination of the following component problems:

(I)

The homogeneous linear IVP:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{U}_t=U_1,\hfill & x\in \mathbb{R},t\in (0,T),\hfill \\ {\left(U_1\right)}_t=U_{xx},\hfill & x\in \mathbb{R},t\in (0,T),\hfill \\ U(x,0)=U_0\left(x\right),U_t(x,0)=U_{*}\left(x\right),\hfill & x\in \mathbb{R}.\hfill \end{array}\right.\end{array}$$

(29)

Using the Fourier transform, we derive the following solutions:

$$\begin{array}{cc}\hfill U\left(x,t\right)& =\widetilde{S_1}\left[U_0,U_{*};0\right]\left(x,t\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{\infty }{\displaystyle \frac{e^{ikx}}{2i\left|k\right|}}\left[\left(e^{-i\left|k\right|t}+e^{i\left|k\right|t}\right)\left(i\left|k\right|\widehat{U_0}\right)+\left(e^{i\left|k\right|t}-e^{-i\left|k\right|t}\right)\widehat{U_{*}}\right]dk,\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill U_1\left(x,t\right)& =\widetilde{S_2}\left[U_0,U_{*};0\right]\left(x,t\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{4\pi }}{\int }_{-\infty }^{\infty }{e}^{ikx}\left[\left(e^{i\left|k\right|t}-e^{-i\left|k\right|t}\right)\left(i\left|k\right|\widehat{U_0}\right)+\left(e^{i\left|k\right|t}+e^{-i\left|k\right|t}\right)\widehat{U_{*}}\right]dk,\hfill \end{array}$$

(31)

where

$$\begin{array}{c}\hfill \widehat{U_0}\left(\xi \right)={\int }_{x\in \mathbb{R}}{e}^{-i\xi x}{U}_0\left(x\right)dx,\widehat{U_{*}}\left(\xi \right)={\int }_{x\in \mathbb{R}}{e}^{-i\xi x}{U}_{*}\left(x\right)dx.\end{array}$$

(II)

The forced linear IVP with zero initial condition:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{W}_t=W_1,\hfill & x\in \mathbb{R},t\in (0,T),\hfill \\ {\left(W_1\right)}_t=W_{xx}+F(x,t),\hfill & x\in \mathbb{R},t\in (0,T),\hfill \\ W(x,0)=W_t(x,0)=0,\hfill & x\in \mathbb{R}.\hfill \end{array}\right.\end{array}$$

(32)

Using the Fourier transform, we derive the following solutions:

$$\begin{array}{cc}\hfill W\left(x,t\right)& =\widetilde{S_1}[0,0;F]\left(x,t\right)={\displaystyle \frac{1}{2\pi }}{\int }_0^t{\int }_{-\infty }^{\infty }{\displaystyle \frac{e^{ikx}}{2i\left|k\right|}}\widehat{F}\left(k,s\right)\left(e^{i\left|k\right|\left(t-s\right)}-e^{-i\left|k\right|\left(t-s\right)}\right)dkds,\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill W_1\left(x,t\right)& =\widetilde{S_2}[0,0;F]\left(x,t\right)={\displaystyle \frac{1}{4\pi }}{\int }_0^t{\int }_{-\infty }^{\infty }{e}^{ikx}\widehat{F}\left(k,s\right)\left(e^{i\left|k\right|\left(t-s\right)}+e^{-i\left|k\right|\left(t-s\right)}\right)dkds,\hfill \end{array}$$

(34)

where

$$\begin{array}{c}\hfill \widehat{W}(\xi ,t)={\int }_{x\in \mathbb{R}}{e}^{-ix\xi }W(x,t)dx.\end{array}$$

(III)

The linear IBVP on the half-line:

$$\begin{array}{c}\hfill \left\{\begin{array}{ccc}\hfill {\left(\tilde{u}\right)}_t& =\widetilde{u_1},\hfill & x\in (0,\infty ),t\in (0,T),\hfill \\ \hfill {\left(\widetilde{u_1}\right)}_t& ={\tilde{u}}_{xx},\hfill & x\in (0,\infty ),t\in (0,T),\hfill \\ \hfill \tilde{u}(x,0)={\tilde{u}}_t(x,0)& =0,\hfill & x\in [0,\infty ),\hfill \\ \hfill {\tilde{u}}_x(0,t)-{\varpi }_1\tilde{u}(0,t)& ={\gamma }_1\left(t\right)-U_x(0,t)-W_x(0,t)\hfill & \\ \hfill & \doteq G_0\left(t\right),\hfill & t\in [0,T],\hfill \end{array}\right.\end{array}$$

(35)

where ${\varpi }_1\ge 1$, and with the solution

$$\begin{array}{c}\hfill \tilde{u}(x,t)=S_1[0,0,G_0;0](x,t)\end{array}$$

(36)

(IV)

The homogeneous linear IBVP with zero initial condition:

$$\begin{array}{c}\hfill \left\{\begin{array}{ccc}\hfill {\left(u^{*}\right)}_t& =u_1^{*},\hfill & x\in (0,\infty ),t\in (0,T),\hfill \\ \hfill {\left(u_1^{*}\right)}_t& =u_{xx}^{*},\hfill & x\in (0,\infty ),t\in (0,T),\hfill \\ \hfill u^{*}(x,0)=u_t^{*}(x,0)& =0,\hfill & x\in [0,\infty ),t\in (0,T),\hfill \\ \hfill u_x^{*}(0,t)-{\varpi }_1{u}^{*}(0,t)& ={\varpi }_{1}U(0,t)+{\varpi }_{1}W(0,t)\hfill & \\ \hfill & \doteq H_0\left(t\right),\hfill & t\in [0,T],\hfill \end{array}\right.\end{array}$$

(37)

where ${\varpi }_1\ge 1$, and with the solution

$$\begin{array}{c}\hfill u^{*}(x,t)=S_1[0,0,H_0;0](x,t).\end{array}$$

(38)

By utilizing the superposition principle, the UTM solutions (9) and (10) for the linear IBVPs (6) have been represented as

$$\begin{array}{cc}\hfill S_1[u_0,u_{*},{\gamma }_1;f](x,t)& =\widetilde{S_1}[U_0,U_{*};0]{{|}_{x>0}+\widetilde{S_1}[0,0;F]|}_{x>0}\hfill \\ \hfill & +S_1[0,0,G_0;0](x,t)+S_1[0,0,H_0;0](x,t),\hfill \end{array}$$

(39)

where the four terms on the right-hand side of (39) correspond to the solutions of the respective problems (29), (32), (35) and (37), respectively.

4.2. The Estimations for the Linear IVPs in Sobolev Spaces

In this subsection, we will derive the space and time estimates for the components of (39), namely $\widetilde{S_1}[U_0,U_{*};0]{|}_{x>0}$, $\widetilde{S_1}{[0,0;F]|}_{x>0}$, $S_1[0,0,G_0;0](x,t)$, and $S_1[0,0,H_0;0](x,t)$.

Theorem 7

(Estimates for Homogeneous IVP (29)). The solutions $U=\widetilde{S_1}[U_0,U_{*};0]$ and $U_1=\widetilde{S_2}[U_0,U_{*};0]$ of the linear IVP (29) given by Formulas (30) and (31) admit the estimates

$$\begin{array}{cc}\hfill \underset{t\in [0,T]}{sup}{\Vert U\left(t\right)\Vert }_{H_x^s\left(\mathbb{R}\right)}& \le {\left(C_2\right)}_s(\Vert U_0{\Vert }_{H_x^s\left(\mathbb{R}\right)}+\Vert U_{*}{\Vert }_{H_x^s\left(\mathbb{R}\right)}),s\in \mathbb{R},\hfill \end{array}$$

(40)

$$\begin{array}{cc}\hfill \underset{t\in [0,T]}{sup}{\Vert U_1\left(t\right)\Vert }_{H_x^{s-1}\left(\mathbb{R}\right)}& \le {\left(C_2\right)}_s(\Vert U_0{\Vert }_{H_x^s\left(\mathbb{R}\right)}+\Vert U_{*}{\Vert }_{H_x^s\left(\mathbb{R}\right)}),s\in \mathbb{R},\hfill \end{array}$$

(41)

$$\begin{array}{cc}\hfill \underset{x\in [0,\infty )}{sup}{\Vert U\left(x\right)\Vert }_{H_t^s(0,T)}& \le {\left(C_2\right)}_s(\Vert U_0{\Vert }_{H_x^s\left(\mathbb{R}\right)}+\Vert U_{*}{\Vert }_{H_x^s\left(\mathbb{R}\right)}),s>0,\hfill \end{array}$$

(42)

$$\begin{array}{cc}\hfill \underset{x\in [0,\infty )}{sup}{\Vert U_1\left(x\right)\Vert }_{H_t^{s-1}(0,T)}& \le {\left(C_2\right)}_s(\Vert U_0{\Vert }_{H_x^s\left(\mathbb{R}\right)}+\Vert U_{*}{\Vert }_{H_x^s\left(\mathbb{R}\right)}),s>1,\hfill \end{array}$$

(43)

where ${\left(C_2\right)}_s>0$ is a constant depending on s.

Proof. 

First, we will prove (40). We will analyze the solution formula given in (30):

$$\begin{array}{cc}\hfill U\left(x,t\right)& =\widetilde{S_1}\left[U_0,U_{*};0\right]\left(x,t\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{\infty }{\displaystyle \frac{e^{ikx}}{2i\left|k\right|}}\left[\left(e^{-i\left|k\right|t}+e^{i\left|k\right|t}\right)\left(i\left|k\right|\widehat{U_0}\left(k\right)\right)+\left(e^{i\left|k\right|t}-e^{-i\left|k\right|t}\right)\widehat{U_{*}}\left(k\right)\right]dk\hfill \\ \hfill & ={\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^0{\displaystyle \frac{e^{ikx}}{-2ik}}\left[\left(e^{ikt}+e^{-ikt}\right)\left(-ik\widehat{U_0}\left(k\right)\right)+\left(e^{-ikt}-e^{ikt}\right)\widehat{U_{*}}\left(k\right)\right]dk\hfill \\ \hfill & +{\displaystyle \frac{1}{2\pi }}{\int }_0^{\infty }{\displaystyle \frac{e^{ikx}}{2ik}}\left[\left(e^{-ikt}+e^{ikt}\right)\left(ik\widehat{U_0}\left(k\right)\right)+\left(e^{ikt}-e^{-ikt}\right)\widehat{U_{*}}\left(k\right)\right]dk.\hfill \end{array}$$

Now, we will calculate the estimate for ${\parallel U\left(t\right)\parallel }_{H_x^s\left(\mathbb{R}\right)}^2$:

$$\begin{array}{cc}\hfill & \Vert U\left(t\right){\Vert }_{H_x^s\left(\mathbb{R}\right)}^2\hfill \\ \hfill & \lesssim \Vert {\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^0{e}^{ikx}\left({\displaystyle \frac{1}{-2ik}}\left(\left(e^{ikt}+e^{-ikt}\right)\left(-ik\widehat{U_0}\left(k\right)\right)+\left(e^{-ikt}-e^{ikt}\right)\widehat{U_{*}}\left(k\right)\right)\right){dk\Vert }_{H_x^s\left(\mathbb{R}\right)}^2\hfill \\ \hfill & +\Vert {\displaystyle \frac{1}{2\pi }}{\int }_0^{\infty }{e}^{ikx}\left({\displaystyle \frac{1}{2ik}}\left(\left(e^{-ikt}+e^{ikt}\right)\left(ik\widehat{U_0}\left(k\right)\right)+\left(e^{ikt}-e^{-ikt}\right)\widehat{U_{*}}\left(k\right)\right)\right){dk\Vert }_{H_x^s\left(\mathbb{R}\right)}^2\hfill \\ \hfill & \le {\int }_{\mathbb{R}}{\left(1+k^2\right)}^s{\left|{\displaystyle \frac{1}{-2ik}}\left(\left(e^{ikt}+e^{-ikt}\right)\left(-ik\widehat{U_0}\left(k\right)\right)+\left(e^{-ikt}-e^{ikt}\right)\widehat{U_{*}}\left(k\right)\right)\right|}^{2}dk\hfill \\ \hfill & +{\int }_{\mathbb{R}}{\left(1+k^2\right)}^s{\left|{\displaystyle \frac{1}{2ik}}\left(\left(e^{-ikt}+e^{ikt}\right)\left(ik\widehat{U_0}\left(k\right)\right)+\left(e^{ikt}-e^{-ikt}\right)\widehat{U_{*}}\left(k\right)\right)\right|}^{2}dk\hfill \\ \hfill & (\mathrm{by}\mathrm{Theorem} 3)\hfill \\ \hfill & \lesssim \underset{\left(A\right)}{\underbrace{{\int }_{\mathbb{R}}{\left(1+k^2\right)}^s{\left|\left(e^{ikt}+e^{-ikt}\right)\widehat{U_0}\left(k\right)\right|}^{2}dk}}+\underset{\left(B\right)}{\underbrace{{\int }_{\mathbb{R}}{\left(1+k^2\right)}^s{\left|{\displaystyle \frac{\left(e^{-ikt}-e^{ikt}\right)}{-2ik}}\widehat{U_{*}}\left(k\right)\right|}^{2}dk}},\hfill \end{array}$$

where

$$\begin{array}{cc}\hfill \left(A\right)& ={\int }_{\mathbb{R}}{\left(1+k^2\right)}^s{\left|\left(e^{ikt}+e^{-ikt}\right)\widehat{U_0}\left(k\right)\right|}^{2}dk\lesssim {\int }_{\mathbb{R}}{\left(1+k^2\right)}^s{\left|\widehat{U_0}\left(k\right)\right|}^{2}dk={\Vert U_0\Vert }_{H_x^s\left(\mathbb{R}\right)}^2,\hfill \\ \hfill \left(B\right)& ={\int }_{\mathbb{R}}{\left(1+k^2\right)}^s{\left|{\displaystyle \frac{\left(e^{-ikt}-e^{ikt}\right)}{-2ik}}\widehat{U_{*}}\left(k\right)\right|}^{2}dk={\int }_{\mathbb{R}}{\left(1+k^2\right)}^s{\left|{\displaystyle \frac{-2isin\left(kt\right)}{-2ik}}\widehat{U_{*}}\left(k\right)\right|}^{2}dk\hfill \\ \hfill & \lesssim {\int }_{\mathbb{R}}{\left(1+k^2\right)}^s\left|{\displaystyle \frac{kt}{k}}\right|{\left|\widehat{U_{*}}\left(k\right)\right|}^{2}dk\lesssim {\Vert U_{*}\Vert }_{H_x^s\left(\mathbb{R}\right)}^2,\left(\mathrm{as}t\in [0,T],T<1\right).\hfill \end{array}$$

Thus, we can conclude that

$$\begin{array}{c}\hfill {\Vert U\left(t\right)\Vert }_{H_x^s\left(\mathbb{R}\right)}^2\lesssim \Vert U_0{\Vert }_{H_x^s\left(\mathbb{R}\right)}^2+{\Vert U_{*}\Vert }_{H_x^s\left(\mathbb{R}\right)}^2,\forall t\in [0,T],T<1,\end{array}$$

and then, we have the inequality (40):

$$\begin{array}{c}\hfill \underset{t\in [0,T]}{sup}{\Vert U\left(t\right)\Vert }_{H_x^s\left(\mathbb{R}\right)}\le {\left(C_2\right)}_s(\Vert U_0{\Vert }_{H_x^s\left(\mathbb{R}\right)}+\Vert U_{*}{\Vert }_{H_x^s\left(\mathbb{R}\right)}),s\in \mathbb{R}.\end{array}$$

Second, we proceed with the proof of (41). We examine the solution Formula (31):

$$\begin{array}{cc}\hfill U_1\left(x,t\right)& =\widetilde{S_2}\left[U_0,U_{*};0\right]\left(x,t\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{4\pi }}{\int }_{-\infty }^{\infty }{e}^{ikx}\left(\left(e^{i\left|k\right|t}-e^{-i\left|k\right|t}\right)\left(i\left|k\right|\widehat{U_0}\left(k\right)\right)+\left(e^{i\left|k\right|t}+e^{-i\left|k\right|t}\right)\widehat{U_{*}}\left(k\right)\right)dk\hfill \\ \hfill & =\underset{\left(C\right)}{\underbrace{{\displaystyle \frac{1}{4\pi }}{\int }_{-\infty }^0{e}^{ikx}\left(\left(e^{-ikt}-e^{ikt}\right)\left(-ik\widehat{U_0}\left(k\right)\right)+\left(e^{-ikt}+e^{ikt}\right)\widehat{U_{*}}\left(k\right)\right)dk}}\hfill \\ \hfill & +\underset{\left(D\right)}{\underbrace{{\displaystyle \frac{1}{4\pi }}{\int }_0^{\infty }{e}^{ikx}\left(\left(e^{ikt}-e^{-ikt}\right)\left(ik\widehat{U_0}\left(k\right)\right)+\left(e^{ikt}+e^{-ikt}\right)\widehat{U_{*}}\left(k\right)\right)dk}},\hfill \end{array}$$

then

$$\begin{array}{c}\hfill \Vert U_1{\left(t\right)\Vert }_{H_x^{s-1}\left(\mathbb{R}\right)}^2\lesssim {\Vert \left(C\right)\Vert }_{H_x^{s-1}\left(\mathbb{R}\right)}^2+{\Vert \left(D\right)\Vert }_{H_x^{s-1}\left(\mathbb{R}\right)}^2.\end{array}$$

At this point, we calculate Equation $\left(C\right)$ to derive the following inequality:

$$\begin{array}{cc}\hfill & \Vert \left(C\right){\Vert }_{H_x^{s-1}\left(\mathbb{R}\right)}^2\hfill \\ \hfill & =\Vert {\displaystyle \frac{1}{4\pi }}{\int }_{-\infty }^0{e}^{ikx}\left(\left(e^{-ikt}-e^{ikt}\right)\left(-ik\widehat{U_0}\left(k\right)\right)+\left(e^{-ikt}+e^{ikt}\right)\widehat{U_{*}}\left(k\right)\right){dk\Vert }_{H_x^{s-1}\left(\mathbb{R}\right)}^2\hfill \\ \hfill & \le {\int }_{\mathbb{R}}{\left(1+k^2\right)}^{s-1}{\left|\left(e^{-ikt}-e^{ikt}\right)\left(-ik\widehat{U_0}\left(k\right)\right)+\left(e^{-ikt}+e^{ikt}\right)\widehat{U_{*}}\left(k\right)\right|}^{2}dk,\left(\mathrm{by}\mathrm{Theroem}\right)\hfill \\ \hfill & \lesssim {\int }_{\mathbb{R}}{\left(1+k^2\right)}^{s-1}\left(1+k^2\right){\left|\widehat{U_0}\left(k\right)\right|}^{2}dk+{\int }_{\mathbb{R}}{\left(1+k^2\right)}^{s-1}{\left|\widehat{U_{*}}\left(k\right)\right|}^{2}dk\hfill \\ \hfill & \le {\int }_{\mathbb{R}}{\left(1+k^2\right)}^s{\left|\widehat{U_0}\left(k\right)\right|}^{2}dk+{\int }_{\mathbb{R}}{\left(1+k^2\right)}^s{\left|\widehat{U_{*}}\left(k\right)\right|}^{2}dk=\Vert U_0{\Vert }_{H_x^s\left(\mathbb{R}\right)}^2+{\Vert U_{*}\Vert }_{H_x^s\left(\mathbb{R}\right)}^2.\hfill \end{array}$$

(44)

Following a similar proof process as used for the estimates of $\left(C\right)$ in (44), we also derive the results:

$$\begin{array}{c}\hfill {\Vert \left(D\right)\Vert }_{H_x^{s-1}\left(\mathbb{R}\right)}^2\lesssim \Vert U_0{\Vert }_{H_x^s\left(\mathbb{R}\right)}^2+{\Vert U_{*}\Vert }_{H_x^s\left(\mathbb{R}\right)}^2.\end{array}$$

(45)

By combining Formulas (44) and (45), we arrive at the inequality (41):

$$\begin{array}{c}\hfill \underset{t\in [0,T]}{sup}{\Vert U_1\left(t\right)\Vert }_{H_x^{s-1}\left(\mathbb{R}\right)}\le {\left(C_2\right)}_s\left(\Vert U_0{\Vert }_{H_x^s\left(\mathbb{R}\right)}+{\Vert U_{*}\Vert }_{H_x^s\left(\mathbb{R}\right)}\right),s\in \mathbb{R}.\end{array}$$

Third, we start with the proof of (42). We examine the solution Formula (30):

$$\begin{array}{cc}\hfill U\left(x,t\right)& =\widetilde{S_1}\left[U_0,U_{*};0\right]\left(x,t\right)\hfill \\ \hfill & ={\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{\infty }{\displaystyle \frac{e^{ikx}}{2i\left|k\right|}}\left(\left(e^{-i\left|k\right|t}+e^{i\left|k\right|t}\right)\left(i\left|k\right|\widehat{U_0}\left(k\right)\right)+\left(e^{i\left|k\right|t}-e^{-i\left|k\right|t}\right)\widehat{U_{*}}\left(k\right)\right)dk\hfill \\ \hfill & ={\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{\infty }{\displaystyle \frac{e^{ikx}}{2i\left|k\right|}}\left(2cos\left(\left|k\right|t\right)\left(i\left|k\right|\widehat{U_0}\left(k\right)\right)+\left(2isin\left(\left|k\right|t\right)\right)\widehat{U_{*}}\left(k\right)\right)dk\hfill \\ \hfill & ={\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{\infty }{e}^{ikx}\left(cos\left(\left|k\right|t\right)\widehat{U_0}\left(k\right)+{\displaystyle \frac{sin\left(\left|k\right|t\right)}{\left|k\right|}}\widehat{U_{*}}\left(k\right)\right)dk.\hfill \end{array}$$

For the time estimate, we express $U(x,t)$ as

$$\begin{array}{c}\hfill U(x,t)=I_1(x,t)+I_2(x,t),\end{array}$$

(46)

where

$$\begin{array}{cc}\hfill I_1\left(x,t\right)& ={\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{\infty }{e}^{ikx}Q\left(k\right)\left(cos\left(\left|k\right|t\right)\widehat{U_0}\left(k\right)+{\displaystyle \frac{sin\left(\left|k\right|t\right)}{\left|k\right|}}\widehat{U_{*}}\left(k\right)\right)dk,\hfill \\ \hfill I_2\left(x,t\right)& ={\displaystyle \frac{1}{4\pi }}{\int }_{-\infty }^{\infty }{e}^{ikx}\left(1-Q\left(k\right)\right)\left(\left(e^{-i\left|k\right|t}+e^{i\left|k\right|t}\right)\widehat{U_0}\left(k\right)+{\displaystyle \frac{\left(e^{i\left|k\right|t}-e^{-i\left|k\right|t}\right)\widehat{U_{*}}\left(k\right)}{ik}}\right)dk,\hfill \end{array}$$

and $Q\left(k\right)\in C_0^{\infty }\left(\mathbb{R}\right)$ is a smooth cut-off function, which is defined by

$$\begin{array}{c}\hfill Q\left(k\right)=\left\{\begin{array}{cc}1,\hfill & \mathrm{if}\left|k\right|\le 1,\hfill \\ 0,\hfill & \mathrm{if}\left|k\right|\ge 2,\hfill \end{array}\right.\mathrm{and}0\le Q\left(k\right)\le 1.\end{array}$$

Now, we will calculate the equations $I_1\left(x,t\right)$ and $I_2\left(x,t\right)$ in the following parts $\left(\mathcal{A}\right)$ and $\left(\mathcal{B}\right)$, respectively.

$\left(\mathcal{A}\right)$ The estimation of $I_1$:

For $\mu \in {\mathbb{N}}_0$, by Remark 2, the definition for $\Vert I_1{\left(x\right)\Vert }_{H_t^{\mu }(0,T)}$ is

$$\begin{array}{c}\hfill \Vert I_1{\left(x\right)\Vert }_{H_t^{\mu }(0,T)}^2=\sum _{j=0}^{\mu }{\Vert {\partial }_t^j{I}_1\left(x\right)\Vert }_{L_t^2(0,T)}^2,\end{array}$$

where ${\mathbb{N}}_0=\mathbb{N}\cup \left\{0\right\}$, and $\mathbb{N}$ is the set of natural numbers. For $j\in \mathbb{N}$, by the definition of $Q\left(k\right)$,

$$\begin{array}{cc}\hfill |{\partial }_t^j{I}_1\left(x,t\right)|& \le {\displaystyle \frac{1}{2\pi }}{\int }_{-2}^2{\left|k\right|}^{j}Q\left(k\right)\left|\widehat{U_0}\left(k\right)\right|dk+{\displaystyle \frac{1}{2\pi }}{\int }_{-2}^2{\left|k\right|}^{j-1}Q\left(k\right)\left|\widehat{U_{*}}\left(k\right)\right|dk\hfill \\ \hfill & \le {\displaystyle \frac{2^{j-1}}{\pi }}{\int }_{-2}^2\left|\widehat{U_0}\left(k\right)\right|dk+{\displaystyle \frac{2^{j-2}}{\pi }}{\int }_{-2}^2\left|\widehat{U_{*}}\left(k\right)\right|dk\hfill \\ \hfill & \le 2^{j-1}{\left({\int }_{-2}^2{\left(1+k^2\right)}^{-s}dk\right)}^{{\displaystyle \frac{1}{2}}}{\left({\int }_{\mathbb{R}}{\left(1+k^2\right)}^s{\left|\widehat{U_0}\left(k\right)\right|}^{2}dk\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & +2^{j-2}{\left({\int }_{-2}^2{\left(1+k^2\right)}^{-s}dk\right)}^{{\displaystyle \frac{1}{2}}}{\left({\int }_{\mathbb{R}}{\left(1+k^2\right)}^s{\left|\widehat{U_{*}}\left(k\right)\right|}^{2}dk\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & \lesssim 2^{j-1}\Vert U_0{\Vert }_{H_x^s\left(\mathbb{R}\right)}+2^{j-2}{\Vert U_{*}\Vert }_{H_x^s\left(\mathbb{R}\right)}.\hfill \end{array}$$

(47)

Hence, for any $j\in \mathbb{N}$, by (47), we derive the following two inequalities:

$$\begin{array}{cc}\hfill \Vert {\partial }_t^j{I}_1\left(x,t\right){\Vert }_{L_t^2\left(0,T\right)}& ={\left({\int }_0^T{\left|{\partial }_t^j{I}_1\left(x,t\right)\right|}^{2}dt\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & \lesssim {\left({\int }_0^T{\left(2^{j-1}\Vert U_0{\Vert }_{H_x^s\left(\mathbb{R}\right)}+2^{j-2}{\Vert U_{*}\Vert }_{H_x^s\left(\mathbb{R}\right)}\right)}^{2}dt\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & \lesssim {\left({\int }_0^T{2}^{2j-2}\Vert U_0{\Vert }_{H_x^s\left(\mathbb{R}\right)}^{2}dt+{\int }_0^T{2}^{2j-4}{\Vert U_{*}\Vert }_{H_x^s\left(\mathbb{R}\right)}^{2}dt\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & \le 2^{j-1}{\left({\int }_0^T{\Vert U_0\Vert }_{H_x^s\left(\mathbb{R}\right)}^{2}dt\right)}^{{\displaystyle \frac{1}{2}}}+2^{j-2}{\left({\int }_0^T{\Vert U_{*}\Vert }_{H_x^s\left(\mathbb{R}\right)}^{2}dt\right)}^{{\displaystyle \frac{1}{2}}}\hfill \\ \hfill & \le 2^{j-1}\sqrt{T}\left(\Vert U_0{\Vert }_{H_x^s\left(\mathbb{R}\right)}+{\Vert U_{*}\Vert }_{H_x^s\left(\mathbb{R}\right)}\right),\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \Vert I_1\left(x,t\right){\Vert }_{L_t^2\left(0,T\right)}^2& ={\int }_0^T{\left|I_1\left(x,t\right)\right|}^{2}dt\hfill \\ \hfill & ={\int }_0^T{\left|{\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{\infty }{e}^{ikx}Q\left(k\right)\left(cos\left(\left|k\right|t\right)\widehat{U_0}\left(k\right)+{\displaystyle \frac{sin\left(\left|k\right|t\right)}{\left|k\right|}}\widehat{U_{*}}\left(k\right)\right)dk\right|}^{2}dt\hfill \\ \hfill & \le {\int }_0^T{\left({\int }_{-2}^2\left(\left|\widehat{U_0}\left(k\right)\right|+\left|\widehat{U_{*}}\left(k\right)\right|\right)dk\right)}^{2}dt\hfill \\ \hfill & \lesssim {\int }_0^T{\left({\int }_{-2}^2\left|\widehat{U_0}\left(k\right)\right|dk\right)}^{2}dt+{\int }_0^T{\left({\int }_{-2}^2\left|\widehat{U_{*}}\left(k\right)\right|dk\right)}^{2}dt\hfill \\ \hfill & \le {\int }_0^T\left({\int }_{-2}^2{\left(1+k^2\right)}^{-s}dk\right)\left({\int }_{\mathbb{R}}{\left(1+k^2\right)}^s{\left|\widehat{U_0}\left(k\right)\right|}^{2}dk\right)dt\hfill \\ \hfill & +{\int }_0^T\left({\int }_{-2}^2{\left(1+k^2\right)}^{-s}dk\right)\left({\int }_{\mathbb{R}}{\left(1+k^2\right)}^s{\left|\widehat{U_{*}}\left(k\right)\right|}^{2}dk\right)dt\hfill \\ \hfill & \lesssim \left(\Vert U_0{\Vert }_{H_x^s\left(\mathbb{R}\right)}^2+{\Vert U_{*}\Vert }_{H_x^s\left(\mathbb{R}\right)}^2\right),\left(\mathrm{since}{\int }_{-2}^2{\left(1+k^2\right)}^{-s}dk\mathrm{is}\mathrm{finite}\mathrm{and}s>0\right).\hfill \end{array}$$

Thus, we have the following two inequalities:

$$\begin{array}{cc}\hfill \Vert {\partial }_t^j{I}_1\left(x,t\right){\Vert }_{L_t^2\left(0,T\right)}& \lesssim 2^{j-1}\sqrt{T}\left(\Vert U_0{\Vert }_{H_x^s\left(\mathbb{R}\right)}+{\Vert U_{*}\Vert }_{H_x^s\left(\mathbb{R}\right)}\right),\hfill \end{array}$$

(48)

$$\begin{array}{cc}\hfill \Vert I_1{(x,t)\Vert }_{L_t^2(0,T)}& \lesssim \Vert U_0{\Vert }_{H_x^s\left(\mathbb{R}\right)}+{\Vert U_{*}\Vert }_{H_x^s\left(\mathbb{R}\right)}.\hfill \end{array}$$

(49)

For $\mu \in \mathbb{N}$, by Formulas (48) and (49), we derive the following inequality:

$$\begin{array}{cc}\hfill \Vert I_1{\left(x\right)\Vert }_{H_t^{\mu }(0,T)}& =\Vert I_1{\left(x\right)\Vert }_{L_t^2(0,T)}+\sum _{j=1}^{\mu }{\Vert {\partial }_t^j{I}_1\left(x\right)\Vert }_{L_t^2(0,T)}\hfill \\ \hfill & \lesssim \Vert U_0{\Vert }_{H_x^s\left(\mathbb{R}\right)}+\Vert U_{*}{\Vert }_{H_x^s\left(\mathbb{R}\right)}+\sum _{j=1}^{\mu }{2}^{j-1}\sqrt{T}(\Vert U_0{\Vert }_{H_x^s\left(\mathbb{R}\right)}+\Vert U_{*}{\Vert }_{H_x^s\left(\mathbb{R}\right)})\hfill \\ \hfill & \lesssim \Vert U_0{\Vert }_{H_x^s\left(\mathbb{R}\right)}+{\Vert U_{*}\Vert }_{H_x^s\left(\mathbb{R}\right)}.\hfill \end{array}$$

Thus, for $\mu \in \mathbb{N}$, we have the following inequality:

$$\begin{array}{c}\hfill \Vert I_1{\left(x\right)\Vert }_{H_t^{\mu }(0,T)}\lesssim \Vert U_0{\Vert }_{H_x^s\left(\mathbb{R}\right)}+{\Vert U_{*}\Vert }_{H_x^s\left(\mathbb{R}\right)}.\end{array}$$

(50)

According to Lemma 1, for any $s>0$ and $s\notin \mathbb{N}$, let $m_1=\lfloor s\rfloor$ and $m_2=\lfloor s\rfloor +1$; then, we derive the following inequality:

$$\begin{array}{cc}\hfill \Vert I_1{\Vert }_{H^s(0,T)}& \le \Vert I_1{\Vert }_{H^{\lfloor s\rfloor }(0,T)}^{\lfloor s\rfloor +1-s}{\Vert I_1\Vert }_{H^{\lfloor s\rfloor +1}(0,T)}^{s-\lfloor s\rfloor }\hfill \\ \hfill & \lesssim \Vert U_0{\Vert }_{H_x^s\left(\mathbb{R}\right)}+{\Vert U_{*}\Vert }_{H_x^s\left(\mathbb{R}\right)},(\mathrm{by}\left(49\right)\mathrm{and}\left(50\right)).\hfill \end{array}$$

Thus, we derive the following inequality:

$$\begin{array}{c}\hfill \Vert I_1{\Vert }_{H^s(0,T)}\lesssim \Vert U_0{\Vert }_{H_x^s\left(\mathbb{R}\right)}+{\Vert U_{*}\Vert }_{H_x^s\left(\mathbb{R}\right)},\mathrm{for}s>0.\end{array}$$

(51)

$\left(\mathcal{B}\right)$ The estimation of $I_2$:

Now, we consider $I_2(x,t)$:

$$\begin{array}{cc}\hfill I_2\left(x,t\right)& ={\displaystyle \frac{1}{4\pi }}{\int }_{-\infty }^{\infty }{e}^{ikx}\left(1-Q\left(k\right)\right)\left(\left(e^{-i\left|k\right|t}+e^{i\left|k\right|t}\right)\left(\widehat{U_0}\left(k\right)\right)+{\displaystyle \frac{\left(e^{i\left|k\right|t}-e^{-i\left|k\right|t}\right)\widehat{U_{*}}\left(k\right)}{ik}}\right)dk\hfill \\ \hfill & =I_{21}(x,t)+I_{22}(x,t),\hfill \end{array}$$

(52)

where

$$\begin{array}{cc}\hfill I_{21}(x,t)& ={\displaystyle \frac{1}{4\pi }}{\int }_{\left|k\right|\ge 1}{e}^{ikx+i\left|k\right|t}\left(\widehat{U_0}\left(k\right)+{\displaystyle \frac{\widehat{U_{*}}\left(k\right)}{ik}}\right)dk,\hfill \\ \hfill I_{22}(x,t)& ={\displaystyle \frac{1}{2\pi }}{\int }_{\left|k\right|\ge 1}{e}^{ikx-i\left|k\right|t}\left(\widehat{U_0}\left(k\right)-{\displaystyle \frac{\widehat{U_{*}}\left(k\right)}{ik}}\right)dk.\hfill \end{array}$$

Next, we will calculate $I_{21}(x,t)$ in the following:

$$\begin{array}{cc}\hfill I_{21}\left(x,t\right)& ={\displaystyle \frac{1}{4\pi }}{\int }_{-\infty }^{-1}{e}^{ikx-ikt}\left(\widehat{U_0}\left(k\right)+{\displaystyle \frac{\widehat{U_{*}}\left(k\right)}{ik}}\right)dk+{\displaystyle \frac{1}{4\pi }}{\int }_1^{\infty }{e}^{ikx+ikt}\left(\widehat{U_0}\left(k\right)+{\displaystyle \frac{\widehat{U_{*}}\left(k\right)}{ik}}\right)dk\hfill \\ \hfill & ={\displaystyle \frac{1}{4\pi }}{\int }_1^{\infty }{e}^{-ikx+ikt}\left(\widehat{U_0}\left(-k\right)-{\displaystyle \frac{\widehat{U_{*}}\left(-k\right)}{ik}}\right)dk+{\displaystyle \frac{1}{4\pi }}{\int }_1^{\infty }{e}^{ikx+ikt}\left(\widehat{U_0}\left(k\right)+{\displaystyle \frac{\widehat{U_{*}}\left(k\right)}{ik}}\right)dk\hfill \\ \hfill & ={\displaystyle \frac{1}{4\pi }}{\int }_1^{\infty }{e}^{ikt}\left(e^{-ikx}\left(\widehat{U_0}\left(-k\right)-{\displaystyle \frac{\widehat{U_{*}}\left(-k\right)}{ik}}\right)+e^{ikx}\left(\widehat{U_0}\left(k\right)+{\displaystyle \frac{\widehat{U_{*}}\left(k\right)}{ik}}\right)\right)dk,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \Vert I_{21}\left(x,t\right){\Vert }_{H_t^s\left(0,T\right)}^2& \le {\int }_1^{\infty }{\left(1+k^2\right)}^s{\left|e^{-ikx}\left(\widehat{U_0}\left(-k\right)-{\displaystyle \frac{\widehat{U_{*}}\left(-k\right)}{ik}}\right)+e^{ikx}\left(\widehat{U_0}\left(k\right)+{\displaystyle \frac{\widehat{U_{*}}\left(k\right)}{ik}}\right)\right|}^{2}dk\hfill \\ \hfill & \lesssim {\int }_1^{\infty }{\left(1+k^2\right)}^s{\left|\widehat{U_0}\left(-k\right)\right|}^{2}dk+{\int }_1^{\infty }{\left(1+k^2\right)}^s{\left|{\displaystyle \frac{\widehat{U_{*}}\left(-k\right)}{ik}}\right|}^{2}dk\hfill \\ \hfill & +{\int }_1^{\infty }{\left(1+k^2\right)}^s{\left|\widehat{U_0}\left(k\right)\right|}^{2}dk+{\int }_1^{\infty }{\left(1+k^2\right)}^s{\left|{\displaystyle \frac{\widehat{U_{*}}\left(k\right)}{ik}}\right|}^{2}dk\hfill \\ \hfill & \lesssim {\int }_{\mathbb{R}}{\left(1+k^2\right)}^s{\left|\widehat{U_0}\left(k\right)\right|}^{2}dk+{\int }_{\mathbb{R}}{\left(1+k^2\right)}^s{\left|\widehat{U_{*}}\left(k\right)\right|}^{2}dk\hfill \\ \hfill & =\Vert U_0{\Vert }_{H_x^s\left(\mathbb{R}\right)}^2+{\Vert U_{*}\Vert }_{H_x^s\left(\mathbb{R}\right)}^2,\hfill \end{array}$$

and hence, we have the following inequality:

$$\begin{array}{c}\hfill \Vert I_{21}{(x,t)\Vert }_{H_t^s(0,T)}\lesssim \Vert U_0{\Vert }_{H_x^s\left(\mathbb{R}\right)}+{\Vert U_{*}\Vert }_{H_x^s\left(\mathbb{R}\right)}.\end{array}$$

(53)

By employing a similar proof strategy as in (53), we obtain the following inequality:

$$\begin{array}{c}\hfill \Vert I_{22}{(x,t)\Vert }_{H_t^s(0,T)}\lesssim \Vert U_0{\Vert }_{H_x^s\left(\mathbb{R}\right)}+{\Vert U_{*}\Vert }_{H_x^s\left(\mathbb{R}\right)}.\end{array}$$

(54)

Consequently, we obtain the following inequality:

$$\begin{array}{cc}\hfill {\Vert U\left(x\right)\Vert }_{H_t^s(0,T)}& \le \Vert I_1{(x,t)\Vert }_{H_t^s(0,T)}+{\Vert I_2(x,t)\Vert }_{H_t^s(0,T)},\left(\mathrm{by}\right(46\left)\right)\hfill \\ \hfill & \le \Vert I_1{(x,t)\Vert }_{H_t^s(0,T)}+\Vert I_{21}{(x,t)\Vert }_{H_t^s(0,T)}+{\Vert I_{22}(x,t)\Vert }_{H_t^s(0,T)},\left(\mathrm{by}\right(52\left)\right)\hfill \\ \hfill & \le {\left(C_2\right)}_s(\Vert U_0{\Vert }_{H_x^s\left(\mathbb{R}\right)}+\Vert U_{*}{\Vert }_{H_x^s\left(\mathbb{R}\right)}),\left(\mathrm{by}\right(51),\left(53\right)\mathrm{and}\left(54\right)).\hfill \end{array}$$

As a result, we establish the inequality (42):

$$\begin{array}{c}\hfill \underset{x\in [0,\infty )}{sup}{\Vert U\left(x\right)\Vert }_{H_t^s(0,T)}\le {\left(C_2\right)}_s(\Vert U_0{\Vert }_{H_x^s\left(\mathbb{R}\right)}+\Vert U_{*}{\Vert }_{H_x^s\left(\mathbb{R}\right)}),s>0.\end{array}$$

Ultimately, by employing a similar proof strategy as in (42), we establish (43):

$$\begin{array}{c}\hfill \underset{x\in [0,\infty )}{sup}\Vert U_1{\left(x\right)\Vert }_{H_t^{s-1}(0,T)}\le {\left(C_2\right)}_s(\Vert U_0{\Vert }_{H_x^s\left(\mathbb{R}\right)}+\Vert U_{*}{\Vert }_{H_x^s\left(\mathbb{R}\right)}),s>1.\end{array}$$

We finish this proof of Theorem 7. □

Now, we will show the estimates of W and $W_1$ in the following Theorem.

Theorem 8

(Sobolev-type estimates for the homogeneous linear IVP (32)). The solutions $W=\widetilde{S_1}[0,0;F]$ and $W_1=\widetilde{S_2}[0,0;F]$ of the linear IVP (32) given by Formulas (33) and (34) admit the following estimates:

Space estimates:

$$\begin{array}{cc}\hfill \underset{t\in [0,T]}{sup}{\Vert W\left(t\right)\Vert }_{H_x^s\left(\mathbb{R}\right)}& \lesssim T\underset{t\in [0,T]}{sup}{\Vert F\left(t\right)\Vert }_{H_x^s\left(\mathbb{R}\right)},s\in \mathbb{R},\hfill \end{array}$$

(55)

$$\begin{array}{cc}\hfill \underset{t\in [0,T]}{sup}{\Vert W_1\left(t\right)\Vert }_{H_x^{s-1}\left(\mathbb{R}\right)}& \lesssim T\underset{t\in [0,T]}{sup}{\Vert F\left(t\right)\Vert }_{H_x^s\left(\mathbb{R}\right)},s\in \mathbb{R}.\hfill \end{array}$$

(56)

Time estimates:

$$\begin{array}{cc}\hfill \underset{x\in [0,\infty )}{sup}{\Vert W\left(x\right)\Vert }_{H_t^s(0,T)}& \lesssim \sqrt{T}\underset{t\in [0,T]}{sup}{\Vert F\left(t\right)\Vert }_{H_x^s\left(\mathbb{R}\right)},\mathrm{for}{\displaystyle \frac{1}{2}}<s<1,\hfill \end{array}$$

(57)

$$\begin{array}{cc}\hfill \underset{x\in [0,\infty )}{sup}{\Vert W_1\left(x\right)\Vert }_{H_t^{s-1}(0,T)}& \lesssim \sqrt{T}\underset{t\in [0,T]}{sup}{\Vert F\left(t\right)\Vert }_{H_x^s\left(\mathbb{R}\right)},\mathrm{for}{\displaystyle \frac{3}{2}}<s<2.\hfill \end{array}$$

(58)

Proof. 

Using the Fourier transform, we have the solution formulas:

$$\begin{array}{cc}\hfill W(x,t)=\widetilde{S_1}[0,0;F](x,t)& ={\displaystyle \frac{1}{2\pi }}{\int }_0^t{\int }_{-\infty }^{\infty }{\displaystyle \frac{e^{ikx}}{2i\left|k\right|}}\widehat{F}(k,t^{\prime })(e^{i\left|k\right|(t-t^{\prime })}-e^{-i\left|k\right|(t-t^{\prime })})dkd{t}^{\prime },\hfill \\ \hfill W_1(x,t)=\widetilde{S_2}[0,0;F](x,t)& ={\displaystyle \frac{1}{4\pi }}{\int }_0^t{\int }_{-\infty }^{\infty }{e}^{ikx}\widehat{F}(k,t^{\prime })(e^{i\left|k\right|(t-t^{\prime })}+e^{-i\left|k\right|(t-t^{\prime })})dkd{t}^{\prime }.\hfill \end{array}$$

(59)

First, we begin with the proof of (55). We will focus on estimating ${\Vert W\left(t\right)\Vert }_{H_x^s\left(\mathbb{R}\right)}$:

$$\begin{array}{cc}\hfill \Vert W\left(t\right){\Vert }_{H_x^s\left(\mathbb{R}\right)}^2& =\Vert {\int }_0^t\left({\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{\infty }{\displaystyle \frac{e^{ikx}}{2i\left|k\right|}}\widehat{F}\left(k,t^{\prime }\right)\left(e^{i\left|k\right|\left(t-t^{\prime }\right)}-e^{-i\left|k\right|\left(t-t^{\prime }\right)}\right)dk\right)d{t}^{\prime }{\Vert }_{H_x^s\left(\mathbb{R}\right)}^2\hfill \\ \hfill & =\Vert {\int }_0^t\widetilde{S_1}[0,F;0]\left(x,t-t^{\prime }\right)d{t}^{\prime }{\Vert }_{H_x^s\left(\mathbb{R}\right)}^2\left(\mathrm{by}\left(30\right)\right)\hfill \\ \hfill & \le {\left({\int }_0^t{\Vert \widetilde{S_1}[0,F;0]\left(x,t-t^{\prime }\right)\Vert }_{H_x^s\left(\mathbb{R}\right)}d{t}^{\prime }\right)}^2\hfill \\ \hfill & \le {\left({\int }_0^T\underset{t\in [0,T]}{sup}{\Vert \widetilde{S_1}[0,F;0]\left(x,t\right)\Vert }_{H_x^s\left(\mathbb{R}\right)}d{t}^{\prime }\right)}^2\hfill \\ \hfill & =T^2{\left(\underset{t\in [0,T]}{sup}{\Vert \widetilde{S_1}[0,F;0]\left(t\right)\Vert }_{H_x^s\left(\mathbb{R}\right)}\right)}^2.\hfill \end{array}$$

Consequently, we obtain the following inequality:

$$\begin{array}{c}\hfill \Vert W\left(t\right){\Vert }_{H_x^s\left(\mathbb{R}\right)}\le T\underset{t\in [0,T]}{sup}{\Vert \widetilde{S_1}[0,F;0]\left(t\right)\Vert }_{H_x^s\left(\mathbb{R}\right)},\forall t\in [0,T].\end{array}$$

As a result, we establish the inequality (55):

$$\begin{array}{cc}\hfill \underset{t\in [0,T]}{sup}{\Vert W\left(t\right)\Vert }_{H_x^s\left(\mathbb{R}\right)}& \le T\underset{t\in [0,T]}{sup}{\Vert \widetilde{S_1}[0,F;0]\left(t\right)\Vert }_{H_x^s\left(\mathbb{R}\right)}\hfill \\ \hfill & \lesssim T\underset{t\in [0,T]}{sup}{\Vert F\left(t\right)\Vert }_{H_x^s\left(\mathbb{R}\right)},s\in \mathbb{R}.(\mathrm{by}\left(40\right)).\hfill \end{array}$$

Second, by employing a similar proof strategy as in (55), we can have the result (56):

$$\begin{array}{c}\hfill \underset{t\in [0,T]}{sup}\Vert W_1\left(t\right){\Vert }_{H_x^{s-1}\left(\mathbb{R}\right)}\lesssim T\underset{t\in [0,T]}{sup}{\Vert F\left(t\right)\Vert }_{H_x^s\left(\mathbb{R}\right)},s\in \mathbb{R}.\end{array}$$

Third, we start with the proof of (57). By Remark 2, the definition for $\Vert W\left(x\right){\Vert }_{H_t^s\left(0,T\right)}$ is

$$\begin{array}{c}\hfill \Vert W\left(x\right){\Vert }_{H_t^s\left(0,T\right)}^2=\Vert W\left(x\right){\Vert }_{L_t^2\left(0,T\right)}^2+{\Vert W\left(x\right)\Vert }_s^2,0\le s<1,\end{array}$$

where

$$\begin{array}{c}\hfill \Vert W\left(x\right){\Vert }_s^2={\int }_0^T{\int }_0^{T-t}{\displaystyle \frac{|W\left(x,t+\xi \right)-W\left(x,t\right){|}^2}{{\xi }^{1+2s}}}d\xi dt.\end{array}$$

Now, we will focus on estimating ${\Vert W\left(x\right)\Vert }_{L_t^2\left(0,T\right)}$:

$$\begin{array}{cc}\hfill \Vert W\left(x\right){\Vert }_{L_t^2\left(0,T\right)}^2& ={\int }_0^T{\left|{\displaystyle \frac{1}{2\pi }}{\int }_0^t{\int }_{-\infty }^{\infty }{\displaystyle \frac{e^{ikx}}{2i\left|k\right|}}\widehat{F}\left(k,t^{\prime }\right)\left(e^{i\left|k\right|(t-t^{\prime })}-e^{-i\left|k\right|\left(t-t^{\prime }\right)}\right)dkd{t}^{\prime }\right|}^{2}dt\hfill \\ \hfill & ={\int }_0^T{\left|{\int }_0^t\widetilde{S_1}[0,F;0]\left(x,t-t^{\prime }\right)d{t}^{\prime }\right|}^{2}dt\hfill \\ \hfill & \le {\left({\int }_0^T{\left({\int }_{t^{\prime }}^T{\left|\widetilde{S_1}[0,F;0]\left(x,t-t^{\prime }\right)\right|}^{2}dt\right)}^{{\displaystyle \frac{1}{2}}}d{t}^{\prime }\right)}^2,\left(\mathrm{by}\mathrm{Lemma}3\right)\hfill \\ \hfill & \le {\left({\int }_0^T{\left({\int }_0^T{\left|\widetilde{S_1}[0,F;0]\left(x,\tau \right)\right|}^{2}d\tau \right)}^{{\displaystyle \frac{1}{2}}}d{t}^{\prime }\right)}^2,\left(\mathrm{let}\tau =t-t^{\prime }\right)\hfill \\ \hfill & \le {\left({\int }_0^T{\Vert \widetilde{S_1}[0,F;0]\left(x\right)\Vert }_{H_{\tau }^s\left(0,T\right)}d{t}^{\prime }\right)}^2\hfill \\ \hfill & \le {\left({\int }_0^T\underset{x\in [0,\infty )}{sup}{\Vert \widetilde{S_1}[0,F;0]\left(x\right)\Vert }_{H_{\tau }^s\left(0,T\right)}d{t}^{\prime }\right)}^2\hfill \\ \hfill & \lesssim {\left({\int }_0^T{\Vert F\left(t\right)\Vert }_{H_x^s\left(\mathbb{R}\right)}d{t}^{\prime }\right)}^2,(\mathrm{by} (42\left)\right)\hfill \\ \hfill & \le T^2{\left(\underset{t\in [0,T]}{sup}{\Vert F\left(t\right)\Vert }_{H_x^s\left(\mathbb{R}\right)}\right)}^2.\hfill \end{array}$$

Consequently, we obtain the following inequality:

$$\begin{array}{c}\hfill \Vert W\left(x\right){\Vert }_{L_t^2\left(0,T\right)}\lesssim T\underset{t\in [0,T]}{sup}{\Vert F\left(t\right)\Vert }_{H_x^s\left(\mathbb{R}\right)}.\end{array}$$

(60)

Next, we will focus on estimating $\Vert W\left(x\right){\Vert }_s$. From Equation (59),

$$\begin{array}{cc}\hfill W\left(x,t\right)& =\widetilde{S_1}[0,0;F]\left(x,t\right)={\displaystyle \frac{1}{2\pi }}{\int }_0^t{\int }_{-\infty }^{\infty }{\displaystyle \frac{e^{ikx}}{2i\left|k\right|}}\widehat{F}\left(k,t^{\prime }\right)\left(e^{i\left|k\right|\left(t-t^{\prime }\right)}-e^{-i\left|k\right|\left(t-t^{\prime }\right)}\right)dkd{t}^{\prime }\hfill \\ \hfill & ={\int }_0^t\left({\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{\infty }{\displaystyle \frac{e^{ikx}}{2i\left|k\right|}}\widehat{F}\left(k,t^{\prime }\right)\left(e^{i\left|k\right|\left(t-t^{\prime }\right)}-e^{-i\left|k\right|\left(t-t^{\prime }\right)}\right)dk\right)d{t}^{\prime }\hfill \\ \hfill & ={\int }_0^t\widetilde{S_1}[0,F;0]\left(x,t-t^{\prime }\right)d{t}^{\prime },\hfill \end{array}$$

and then we obtain

$$\begin{array}{cc}\hfill & W\left(x,t+\zeta \right)-W\left(x,t\right)\hfill \\ \hfill & ={\int }_0^{t+\zeta }\widetilde{S_1}[0,F;0]\left(x,t+\zeta -t^{\prime }\right)d{t}^{\prime }-{\int }_0^t\widetilde{S_1}[0,F;0]\left(x,t-t^{\prime }\right)d{t}^{\prime }\hfill \\ \hfill & ={\int }_0^t\left(\widetilde{S_1}[0,F;0]\left(x,t+\zeta -t^{\prime }\right)-\widetilde{S_1}[0,F;0]\left(x,t-t^{\prime }\right)\right)d{t}^{\prime }\hfill \\ \hfill & +{\int }_t^{t+\zeta }\widetilde{S_1}[0,F;0]\left(x,t+\zeta -t^{\prime }\right)d{t}^{\prime }.\hfill \end{array}$$

(61)

From Equation (61), we derive the estimation of $\Vert W\left(x\right){\Vert }_s^2$:

$$\begin{array}{cc}\hfill & \Vert W\left(x\right){\Vert }_s^2={\int }_0^T{\int }_0^{T-t}{\displaystyle \frac{{\left|W\left(x,t+\zeta \right)-W\left(x,t\right)\right|}^2}{{\zeta }^{1+2s}}}d\zeta dt\hfill \\ \hfill & ={\int }_0^T{\int }_0^{T-t}{\displaystyle \frac{1}{{\zeta }^{1+2s}}}\left|{\int }_0^t\left(\widetilde{S_1}[0,F;0]\left(x,t+\zeta -t^{\prime }\right)-\widetilde{S_1}[0,F;0]\left(x,t-t^{\prime }\right)\right)d{t}^{\prime }\right.\hfill \\ \hfill & +{\left.{\int }_t^{t+\zeta }\widetilde{S_1}[0,F;0]\left(x,t+\zeta -t^{\prime }\right)d{t}^{\prime }\right|}^{2}d\zeta dt\hfill \\ \hfill & \lesssim \underset{\left(E\right)}{\underbrace{{\int }_0^T{\int }_0^{T-t}{\displaystyle \frac{1}{{\zeta }^{1+2s}}}{\left|{\int }_0^t\left(\widetilde{S_1}[0,F;0]\left(x,t+\zeta -t^{\prime }\right)-\widetilde{S_1}[0,F;0]\left(x,t-t^{\prime }\right)\right)d{t}^{\prime }\right|}^{2}d\zeta dt}}\hfill \\ \hfill & +\underset{\left(F\right)}{\underbrace{{\int }_0^T{\int }_0^{T-t}{\displaystyle \frac{1}{{\zeta }^{1+2s}}}{\left|{\int }_t^{t+\zeta }\widetilde{S_1}[0,F;0]\left(x,t+\zeta -t^{\prime }\right)d{t}^{\prime }\right|}^{2}d\zeta dt}}.\hfill \end{array}$$

Now, we calculate the equations $\left(E\right)$ and $\left(F\right)$ to derive the following two inequalities:

$$\begin{array}{cc}\hfill \left(E\right)& ={\int }_0^T{\int }_0^{T-t}{\displaystyle \frac{1}{{\zeta }^{1+2s}}}{\left|{\int }_0^t\left(\widetilde{S_1}[0,F;0]\left(x,t+\zeta -t^{\prime }\right)-\widetilde{S_1}[0,F;0]\left(x,t-t^{\prime }\right)\right)d{t}^{\prime }\right|}^{2}d\zeta dt\hfill \\ \hfill & \le {\left({\int }_0^T{\left({\int }_0^T{\int }_0^{T-t}{\displaystyle \frac{1}{{\zeta }^{1+2s}}}{\left|\widetilde{S_1}[0,F;0]\left(x,t+\zeta -t^{\prime }\right)-\widetilde{S_1}[0,F;0]\left(x,t-t^{\prime }\right)\right|}^{2}d\zeta dt\right)}^{{\displaystyle \frac{1}{2}}}d{t}^{\prime }\right)}^2,\hfill \\ \hfill & \left(\mathrm{by}\mathrm{Lemma}3\right)\hfill \\ \hfill & ={\left({\int }_0^T{\Vert \widetilde{S_1}[0,F;0]\left(x,t-t^{\prime }\right)\Vert }_{s}d{t}^{\prime }\right)}^2\le {\left({\int }_0^T{\Vert F\left(t\right)\Vert }_{H_x^s\left(\mathbb{R}\right)}d{t}^{\prime }\right)}^2,(\mathrm{by} (42\left)\right)\hfill \\ \hfill & \le {\left({\int }_0^T\underset{t\in [0,T]}{sup}{\Vert F\left(t\right)\Vert }_{H_x^s\left(\mathbb{R}\right)}d{t}^{\prime }\right)}^2={\left(T\underset{t\in [0,T]}{sup}{\Vert F\left(t\right)\Vert }_{H_x^s\left(\mathbb{R}\right)}\right)}^2,\hfill \end{array}$$

(62)

and

$$\begin{array}{cc}\hfill \left(F\right)& ={\int }_0^T{\int }_0^{T-t}{\displaystyle \frac{1}{{\zeta }^{1+2s}}}{\left|{\int }_t^{t+\zeta }\widetilde{S_1}[0,F;0]\left(x,t+\zeta -t^{\prime }\right)d{t}^{\prime }\right|}^{2}d\zeta dt\hfill \\ \hfill & ={\int }_0^T{\int }_0^{T-t}{\displaystyle \frac{1}{{\zeta }^{1+2s}}}{\left|{\int }_t^{t+\zeta }\left({\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{\infty }{\displaystyle \frac{e^{ikx}}{2i\left|k\right|}}\widehat{F}\left(k,t^{\prime }\right)\left(e^{i\left|k\right|\left(t+\zeta -t^{\prime }\right)}-e^{-i\left|k\right|\left(t+\zeta -t^{\prime }\right)}\right)dk\right)d{t}^{\prime }\right|}^{2}d\zeta dt\hfill \\ \hfill & \le {\int }_0^T{\int }_0^{T-t}{\displaystyle \frac{1}{{\zeta }^{1+2s}}}{\Vert {\displaystyle \frac{1}{2\pi }}{\int }_t^{t+\zeta }{\int }_{-\infty }^{\infty }{\displaystyle \frac{e^{ikx}\widehat{F}\left(k,t^{\prime }\right)\left(e^{i\left|k\right|\left(t+\zeta -t^{\prime }\right)}-e^{-i\left|k\right|\left(t+\zeta -t^{\prime }\right)}\right)}{2i\left|k\right|}}dkd{t}^{\prime }\Vert }_{L_x^{\infty }\left(\mathbb{R}\right)}^{2}d\zeta dt\hfill \\ \hfill & (\mathrm{since}s>{\displaystyle \frac{1}{2}},\mathrm{by}\mathrm{Theorem}5(\mathrm{the}\mathrm{Sobolev}\mathrm{Embedding}\mathrm{Theorem}))\hfill \\ \hfill & \le {\int }_0^T{\int }_0^{T-t}{\displaystyle \frac{1}{{\zeta }^{1+2s}}}{\Vert {\displaystyle \frac{1}{2\pi }}{\int }_t^{t+\zeta }{\int }_{-\infty }^{\infty }{\displaystyle \frac{e^{ikx}\widehat{F}\left(k,t^{\prime }\right)\left(e^{i\left|k\right|\left(t+\zeta -t^{\prime }\right)}-e^{-i\left|k\right|\left(t+\zeta -t^{\prime }\right)}\right)}{2i\left|k\right|}}dkd{t}^{\prime }\Vert }_{H_x^s\left(\mathbb{R}\right)}^{2}d\zeta dt\hfill \\ \hfill & ={\int }_0^T{\int }_0^{T-t}{\displaystyle \frac{1}{{\zeta }^{1+2s}}}{\Vert {\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{\infty }{e}^{ikx}{\displaystyle \frac{{\int }_t^{t+\zeta }\widehat{F}\left(k,t^{\prime }\right)\left(e^{i\left|k\right|\left(t+\zeta -t^{\prime }\right)}-e^{-i\left|k\right|\left(t+\zeta -t^{\prime }\right)}\right)d{t}^{\prime }}{2i\left|k\right|}}dk\Vert }_{H_x^s\left(\mathbb{R}\right)}^{2}d\zeta dt\hfill \\ \hfill & ={\int }_0^T{\int }_0^{T-t}{\displaystyle \frac{1}{{\zeta }^{1+2s}}}\left({\int }_{\mathbb{R}}{\left(1+k^2\right)}^s{\left|{\displaystyle \frac{{\int }_t^{t+\zeta }\widehat{F}\left(k,t^{\prime }\right)\left(e^{i\left|k\right|\left(t+\zeta -t^{\prime }\right)}-e^{-i\left|k\right|\left(t+\zeta -t^{\prime }\right)}\right)d{t}^{\prime }}{2i\left|k\right|}}\right|}^{2}dk\right)d\zeta dt\hfill \\ \hfill & \left(\mathrm{by}\mathrm{Theorem}3\mathrm{and}\mathrm{Theorem}4\right)\hfill \end{array}$$

$$\begin{array}{cc}\hfill & ={\int }_0^T{\int }_0^{T-t}{\displaystyle \frac{1}{{\zeta }^{1+2s}}}\left({\int }_{\mathbb{R}}{\left(1+k^2\right)}^s{\left|{\displaystyle \frac{1}{2i\left|k\right|}}{\int }_t^{t+\zeta }\widehat{F}\left(k,t^{\prime }\right)\left(2isin\left(\left|k\right|\left(t+\zeta -t^{\prime }\right)\right)\right)d{t}^{\prime }\right|}^{2}dk\right)d\zeta dt\hfill \\ \hfill & \le {\int }_0^T{\int }_0^{T-t}{\displaystyle \frac{1}{{\zeta }^{1+2s}}}\left({\int }_{\mathbb{R}}{\left(1+k^2\right)}^s{\left({\displaystyle \frac{1}{2\left|k\right|}}{\int }_t^{t+\zeta }\left|\widehat{F}\left(k,t^{\prime }\right)\right|·2\left|k\right|\left(t+\zeta -t^{\prime }\right)d{t}^{\prime }\right)}^{2}dk\right)d\zeta dt\hfill \\ \hfill & ={\int }_0^T{\int }_0^{T-t}{\displaystyle \frac{1}{{\zeta }^{1+2s}}}\left({\int }_{\mathbb{R}}{\left(1+k^2\right)}^s{\left({\int }_t^{t+\zeta }\left|\widehat{F}\left(k,t^{\prime }\right)\right|\left(t+\zeta -t^{\prime }\right)d{t}^{\prime }\right)}^{2}dk\right)d\zeta dt\hfill \\ \hfill & \le {\int }_0^T{\int }_0^{T-t}{\displaystyle \frac{1}{{\zeta }^{1+2s}}}{\left({\int }_t^{t+\zeta }{\left({\int }_{\mathbb{R}}{\left(1+k^2\right)}^s{\left|\widehat{F}\left(k,t^{\prime }\right)\right|}^2{\left(t+\zeta -t^{\prime }\right)}^{2}dk\right)}^{{\displaystyle \frac{1}{2}}}d{t}^{\prime }\right)}^{2}d\zeta dt\hfill \\ \hfill & \left(\mathrm{by}\mathrm{Lemma}3(\mathrm{Minkowski}\mathrm{integral}\mathrm{inequality})\right)\hfill \\ \hfill & \le {\int }_0^T{\int }_0^{T-t}{\displaystyle \frac{1}{{\zeta }^{1+2s}}}{\left({\int }_t^{t+\zeta }\left(t+\zeta -t^{\prime }\right){\left({\int }_{\mathbb{R}}{\left(1+k^2\right)}^s{\left|\widehat{F}\left(k,t^{\prime }\right)\right|}^{2}dk\right)}^{{\displaystyle \frac{1}{2}}}d{t}^{\prime }\right)}^{2}d\zeta dt\hfill \\ \hfill & ={\int }_0^T{\int }_0^{T-t}{\displaystyle \frac{1}{{\zeta }^{1+2s}}}{\left({\int }_t^{t+\zeta }\left(t+\zeta -t^{\prime }\right){\Vert F\left(t^{\prime }\right)\Vert }_{H_x^s\left(\mathbb{R}\right)}d{t}^{\prime }\right)}^{2}d\zeta dt\hfill \\ \hfill & =\underset{t^{\prime }\in [t,t+\zeta ]}{sup}{\Vert F\left(t^{\prime }\right)\Vert }_{H_x^s\left(\mathbb{R}\right)}^2\left({\int }_0^T{\int }_0^{T-t}{\zeta }^{3-2s}d\zeta dt\right)\hfill \\ \hfill & \lesssim \underset{t^{\prime }\in [t,t+\zeta ]}{sup}\Vert F\left(t^{\prime }\right){\Vert }_{H_x^s\left(\mathbb{R}\right)}^2{T}^{5-2s}\le \underset{t^{\prime }\in [0,T]}{sup}{\Vert F\left(t^{\prime }\right)\Vert }_{H_x^s\left(\mathbb{R}\right)}^{2}T,(\mathrm{for}T<1).\hfill \end{array}$$

Consequently, we obtain the following inequality:

$$\begin{array}{c}\hfill \left(F\right)\lesssim T\underset{t\in [0,T]}{sup}{\Vert F\left(t\right)\Vert }_{H_x^s\left(\mathbb{R}\right)}^2.\end{array}$$

(63)

By (60), (62) and (63), we derive (57):

$$\begin{array}{cc}\hfill & \Vert W\left(x\right){\Vert }_{H_t^s(0,T)}^2\lesssim \left(T\underset{t\in [0,T]}{sup}{\Vert F\left(t\right)\Vert }_{H_x^s\left(\mathbb{R}\right)}^2\right)\hfill \\ \hfill & ⟹\underset{x\in [0,\infty )}{sup}\Vert W\left(x\right){\Vert }_{H_t^s(0,T)}\lesssim \sqrt{T}\underset{t\in [0,T]}{sup}{\Vert F\left(t\right)\Vert }_{H_x^s\left(\mathbb{R}\right)},\mathrm{for}{\displaystyle \frac{1}{2}}<s<1\mathrm{and}0<T<1.\hfill \end{array}$$

Ultimately, by employing a similar proof strategy as in (57), we establish (58):

$$\begin{array}{c}\hfill \underset{x\in [0,\infty )}{sup}\Vert W_1\left(x\right){\Vert }_{H_t^{s-1}(0,T)}\lesssim \sqrt{T}\underset{t\in [0,T]}{sup}{\Vert F\left(t\right)\Vert }_{H_x^s(0,T)},\mathrm{for}{\displaystyle \frac{3}{2}}<s<2\mathrm{and}T<1.\end{array}$$

We finish this proof of Theorem 8. □

4.3. About the Proof of Therorem 1

By applying the superposition principle, we can synthesize Theorem 6, Theorem 7, and Theorem 8 to derive Theorem 1 for the forced linear IVPs (6) and (7). Additionally, we incorporate the following time estimates:

$$\begin{array}{cc}\hfill \Vert G_0{\Vert }_{H_t^{s-1}(0,T)}& =\Vert {\gamma }_1\left(t\right)-U_x\left(0,t\right)-W_x\left(0,t\right){\Vert }_{H_t^{s-1}(0,T)}\hfill \\ \hfill & \le \Vert {\gamma }_1\left(t\right){\Vert }_{H_t^{s-1}(0,T)}+{\Vert U_x\left(0,t\right)\Vert }_{H_t^{s-1}(0,T)}\hfill \\ \hfill & +\Vert W_x\left(0,t\right){\Vert }_{H_t^{s-1}(0,T)}\hfill \end{array}$$

(64)

and

$$\begin{array}{cc}\hfill \Vert H_0{\Vert }_{H_t^{s-1}(0,T)}& =\Vert {\varpi }_{1}U\left(0,t\right)+{\varpi }_{1}W\left(0,t\right){\Vert }_{H_t^{s-1}(0,T)}\hfill \\ \hfill & \le {\varpi }_1\Vert U\left(0,t\right){\Vert }_{H_t^{s-1}(0,T)}+{\varpi }_1{\Vert W\left(0,t\right)\Vert }_{H_t^{s-1}(0,T)}.\hfill \end{array}$$

(65)

Thus, we need to estimate $\Vert U_x{(0,t)\Vert }_{H_t^{s-1}(0,T)}$ and $\Vert W_x{(0,t)\Vert }_{H_t^{s-1}(0,T)}$.

Lemma 4.

(Sobolev-type estimates) For ${\displaystyle \frac{1}{2}}<s<1$, we obtain the following estimates:

$$\begin{array}{c}\hfill \underset{x\in [0,\infty )}{sup}\Vert U_x\left(x\right){\Vert }_{H_t^{s-1}(0,T)}\lesssim \Vert U_0{\Vert }_{H_x^s\left(\mathbb{R}\right)}+{\Vert U_{*}\Vert }_{H_x^s\left(\mathbb{R}\right)}\end{array}$$

(66)

and

$$\begin{array}{c}\hfill \underset{x\in [0,\infty )}{sup}\Vert W_x\left(x\right){\Vert }_{H_t^{s-1}(0,T)}\lesssim \sqrt{T}\underset{t\in [0,T]}{sup}{\Vert F\left(t\right)\Vert }_{H_x^s\left(\mathbb{R}\right)}.\end{array}$$

(67)

Proof. 

First, we begin the proof of (66). From Equation (30), we have

$$\begin{array}{cc}\hfill U\left(x,t\right)& ={\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^0{\displaystyle \frac{e^{ikx}}{-2ik}}\left(\left(e^{ikt}+e^{-ikt}\right)\left(-ik{\widehat{U}}_0\left(k\right)\right)+\left(e^{-ikt}-e^{ikt}\right){\widehat{U}}_{*}\left(k\right)\right)dk\hfill \\ \hfill & +{\displaystyle \frac{1}{2\pi }}{\int }_0^{\infty }{\displaystyle \frac{e^{ikx}}{2ik}}\left(\left(e^{-ikt}+e^{ikt}\right)\left(ik{\widehat{U}}_0\left(k\right)\right)+\left(e^{ikt}-e^{-ikt}\right){\widehat{U}}_{*}\left(k\right)\right)dk.\hfill \end{array}$$

Thus, we obtain that

$$\begin{array}{cc}\hfill U_x\left(x,t\right)& ={\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^0{e}^{ikt}\left({\displaystyle \frac{1}{2}}\left(ik\right)e^{ikx}{\widehat{U}}_0\left(k\right)+{\displaystyle \frac{1}{2}}{e}^{ikx}{\widehat{U}}_{*}\left(k\right)\right)dk\hfill \\ \hfill & +{\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^0{e}^{-ikt}\left({\displaystyle \frac{1}{2}}\left(ik\right)e^{ikx}{\widehat{U}}_0\left(k\right)-{\displaystyle \frac{1}{2}}{e}^{ikx}{\widehat{U}}_{*}\left(k\right)\right)dk\hfill \\ \hfill & +{\displaystyle \frac{1}{2\pi }}{\int }_0^{\infty }{e}^{ikt}\left({\displaystyle \frac{1}{2}}\left(ik\right)e^{ikx}{\widehat{U}}_0\left(k\right)+{\displaystyle \frac{1}{2}}{e}^{ikx}{\widehat{U}}_{*}\left(k\right)\right)dk\hfill \\ \hfill & +{\displaystyle \frac{1}{2\pi }}{\int }_0^{\infty }{e}^{-ikt}\left({\displaystyle \frac{1}{2}}\left(ik\right)e^{ikx}{\widehat{U}}_0\left(k\right)-{\displaystyle \frac{1}{2}}{e}^{ikx}{\widehat{U}}_{*}\left(k\right)\right)dk.\hfill \end{array}$$

According to Theorem 3, the estimation of $\Vert U_x\left(x\right){\Vert }_{H_t^{s-1}(0,T)}^2$ is

$$\begin{array}{cc}\hfill \Vert U_x\left(x\right){\Vert }_{H_t^{s-1}(0,T)}^2& \le \Vert U_x\left(x\right){\Vert }_{H_t^{s-1}\left(\mathbb{R}\right)}^2\hfill \\ \hfill & \lesssim \Vert {\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^0{e}^{ikt}\left({\displaystyle \frac{1}{2}}\left(ik\right)e^{ikx}{\widehat{U}}_0\left(k\right)+{\displaystyle \frac{1}{2}}{e}^{ikx}{\widehat{U}}_{*}\left(k\right)\right){dk\Vert }_{H_t^{s-1}\left(\mathbb{R}\right)}^2\hfill \\ \hfill & +\Vert {\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^0{e}^{-ikt}\left({\displaystyle \frac{1}{2}}\left(ik\right)e^{ikx}{\widehat{U}}_0\left(k\right)-{\displaystyle \frac{1}{2}}{e}^{ikx}{\widehat{U}}_{*}\left(k\right)\right){dk\Vert }_{H_t^{s-1}\left(\mathbb{R}\right)}^2\hfill \\ \hfill & +\Vert {\displaystyle \frac{1}{2\pi }}{\int }_0^{\infty }{e}^{ikt}\left({\displaystyle \frac{1}{2}}ik{e}^{ikx}{\widehat{U}}_0\left(k\right)+{\displaystyle \frac{1}{2}}{e}^{ikx}{\widehat{U}}_{*}\left(k\right)\right){dk\Vert }_{H_t^{s-1}\left(\mathbb{R}\right)}^2\hfill \\ \hfill & +\Vert {\displaystyle \frac{1}{2\pi }}{\int }_0^{\infty }{e}^{-ikt}\left({\displaystyle \frac{1}{2}}ik{e}^{ikx}{\widehat{U}}_0\left(k\right)-{\displaystyle \frac{1}{2}}{e}^{ikx}{\widehat{U}}_{*}\left(k\right)\right){dk\Vert }_{H_t^{s-1}\left(\mathbb{R}\right)}^2\hfill \\ \hfill & \le {\int }_{\mathbb{R}}{\left(1+k^2\right)}^{s-1}{\left|{\displaystyle \frac{1}{2}}\left(ik\right)e^{ikx}{\widehat{U}}_0\left(k\right)+{\displaystyle \frac{1}{2}}{e}^{ikx}{\widehat{U}}_{*}\left(k\right)\right|}^{2}dk\hfill \\ \hfill & +{\int }_{\mathbb{R}}{\left(1+k^2\right)}^{s-1}{\left|-{\displaystyle \frac{1}{2}}ik{e}^{-ikx}{\widehat{U}}_0\left(-k\right)-{\displaystyle \frac{1}{2}}{e}^{-ikx}{\widehat{U}}_{*}\left(-k\right)\right|}^{2}dk\hfill \\ \hfill & +{\int }_{\mathbb{R}}{\left(1+k^2\right)}^{s-1}{\left|{\displaystyle \frac{1}{2}}ik{e}^{ikx}{\widehat{U}}_0\left(k\right)+{\displaystyle \frac{1}{2}}{e}^{ikx}{\widehat{U}}_{*}\left(k\right)\right|}^{2}dk\hfill \\ \hfill & +{\int }_{\mathbb{R}}{\left(1+k^2\right)}^{s-1}{\left|-{\displaystyle \frac{1}{2}}ik{e}^{-ikx}{\widehat{U}}_0\left(-k\right)-{\displaystyle \frac{1}{2}}{e}^{-ikx}{\widehat{U}}_{*}\left(-k\right)\right|}^{2}dk\hfill \\ \hfill & \le {\int }_{\mathbb{R}}{\left(1+k^2\right)}^{s-1}{k}^2{\left|{\widehat{U}}_0\left(k\right)\right|}^{2}dk+{\int }_{\mathbb{R}}{\left(1+k^2\right)}^{s-1}{\left|{\widehat{U}}_{*}\left(k\right)\right|}^{2}dk\hfill \\ \hfill & +{\int }_{\mathbb{R}}{\left(1+k^2\right)}^{s-1}{k}^2{\left|{\widehat{U}}_0\left(-k\right)\right|}^{2}dk+{\int }_{\mathbb{R}}{\left(1+k^2\right)}^{s-1}{\left|{\widehat{U}}_{*}\left(-k\right)\right|}^{2}dk\hfill \\ \hfill & +{\int }_{\mathbb{R}}{\left(1+k^2\right)}^{s-1}{k}^2{\left|{\widehat{U}}_0\left(k\right)\right|}^{2}dk+{\int }_{\mathbb{R}}{\left(1+k^2\right)}^{s-1}{\left|{\widehat{U}}_{*}\left(k\right)\right|}^{2}dk\hfill \\ \hfill & +{\int }_{\mathbb{R}}{\left(1+k^2\right)}^{s-1}{k}^2{\left|{\widehat{U}}_0\left(-k\right)\right|}^{2}dk+{\int }_{\mathbb{R}}{\left(1+k^2\right)}^{s-1}{\left|{\widehat{U}}_{*}\left(-k\right)\right|}^{2}dk\hfill \\ \hfill & \lesssim {\int }_{\mathbb{R}}{\left(1+k^2\right)}^s{\left|{\widehat{U}}_0\left(k\right)\right|}^{2}dk+{\int }_{\mathbb{R}}{\left(1+k^2\right)}^s{\left|{\widehat{U}}_{*}\left(k\right)\right|}^{2}dk\hfill \\ \hfill & =\Vert U_0{\Vert }_{H_x^s\left(\mathbb{R}\right)}^2+{\Vert U_{*}\Vert }_{H_x^s\left(\mathbb{R}\right)}^2.\hfill \end{array}$$

Consequently, we obtain the following inequality:

$$\begin{array}{c}\hfill \Vert U_x\left(x\right){\Vert }_{H_t^{s-1}(0,T)}^2\lesssim \Vert U_0{\Vert }_{H_x^s\left(\mathbb{R}\right)}^2+{\Vert U_{*}\Vert }_{H_x^s\left(\mathbb{R}\right)}^2,\end{array}$$

and then we derive the following inequality (66):

$$\begin{array}{c}\hfill \underset{x\in [0,\infty )}{sup}{\Vert U_x\left(x\right)\Vert }_{H_t^{s-1}(0,T)}\lesssim \left(\Vert U_0{\Vert }_{H_x^s\left(\mathbb{R}\right)}+{\Vert U_{*}\Vert }_{H_x^s\left(\mathbb{R}\right)}\right).\end{array}$$

Next, we begin the proof of (67). We will consider Equation (59):

$$\begin{array}{c}\hfill W\left(x,t\right)=\widetilde{S_1}[0,0;F]\left(x,t\right)={\displaystyle \frac{1}{2\pi }}{\int }_0^t{\int }_{-\infty }^{\infty }{\displaystyle \frac{e^{ikx}}{2i\left|k\right|}}\widehat{F}\left(k,t^{\prime }\right)\left(e^{i\left|k\right|\left(t-t^{\prime }\right)}-e^{-i\left|k\right|\left(t-t^{\prime }\right)}\right)dkd{t}^{\prime },\end{array}$$

and then, we derive that

$$\begin{array}{c}\hfill W_x\left(x,t\right)={\displaystyle \frac{1}{2\pi }}{\int }_0^t{\int }_{-\infty }^{\infty }{\displaystyle \frac{ik}{2i\left|k\right|}}{e}^{ikx}\widehat{F}\left(k,t^{\prime }\right)\left(e^{i\left|k\right|\left(t-t^{\prime }\right)}-e^{-i\left|k\right|\left(t-t^{\prime }\right)}\right)dkd{t}^{\prime }.\end{array}$$

The estimation of $\Vert W_x\left(x,t\right){\Vert }_{H_t^s\left(\mathbb{R}\right)}$ is

$$\begin{array}{cc}\hfill \Vert W_x\left(x,t\right){\Vert }_{H_t^s\left(\mathbb{R}\right)}& \le {\int }_0^T{\Vert {\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^{\infty }{\displaystyle \frac{ik}{2i\left|k\right|}}{e}^{ikx}\widehat{F}\left(k,t^{\prime }\right)\left(e^{i\left|k\right|\left(t-t^{\prime }\right)}-e^{-i\left|k\right|\left(t-t^{\prime }\right)}\right)dk\Vert }_{H_t^s\left(\mathbb{R}\right)}d{t}^{\prime }\hfill \\ \hfill & ={\int }_0^T\Vert {\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^0{\displaystyle \frac{-1}{2}}{e}^{ikx}\widehat{F}\left(k,t^{\prime }\right)\left(e^{i\left|k\right|\left(t-t^{\prime }\right)}-e^{-i\left|k\right|\left(t-t^{\prime }\right)}\right)dk\hfill \\ \hfill & +{\displaystyle \frac{1}{2\pi }}{\int }_0^{\infty }{\displaystyle \frac{1}{2}}{e}^{ikx}\widehat{F}\left(k,t^{\prime }\right)\left(e^{i\left|k\right|\left(t-t^{\prime }\right)}-e^{-i\left|k\right|\left(t-t^{\prime }\right)}\right){dk\Vert }_{H_t^s\left(\mathbb{R}\right)}d{t}^{\prime }\hfill \\ \hfill & \le {\int }_0^T{\Vert {\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^0{e}^{-ikt}{e}^{ikx}{e}^{ik{t}^{\prime }}\widehat{F}\left(k,t^{\prime }\right)dk\Vert }_{H_t^s\left(\mathbb{R}\right)}d{t}^{\prime }\hfill \\ \hfill & +{\int }_0^T{\Vert {\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^0{e}^{ikt}{e}^{ikx}{e}^{-ik{t}^{\prime }}\widehat{F}\left(k,t^{\prime }\right)dk\Vert }_{H_t^s\left(\mathbb{R}\right)}d{t}^{\prime }\hfill \\ \hfill & +{\int }_0^T{\Vert {\displaystyle \frac{1}{2\pi }}{\int }_0^{\infty }{e}^{ikt}{e}^{ikx}{e}^{-ik{t}^{\prime }}\widehat{F}\left(k,t^{\prime }\right)dk\Vert }_{H_t^s\left(\mathbb{R}\right)}d{t}^{\prime }\hfill \\ \hfill & +{\int }_0^T{\Vert {\displaystyle \frac{1}{2\pi }}{\int }_0^{\infty }{e}^{-ikt}{e}^{ikx}{e}^{ik{t}^{\prime }}\widehat{F}\left(k,t^{\prime }\right)dk\Vert }_{H_t^s\left(\mathbb{R}\right)}d{t}^{\prime }\hfill \\ \hfill & ={\int }_0^T{\Vert {\displaystyle \frac{1}{2\pi }}{\int }_0^{\infty }{e}^{ikt}{e}^{-ikx}{e}^{-ik{t}^{\prime }}\widehat{F}\left(-k,t^{\prime }\right)dk\Vert }_{H_t^s\left(\mathbb{R}\right)}d{t}^{\prime }\hfill \\ \hfill & +{\int }_0^T{\Vert {\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^0{e}^{ikt}{e}^{ikx}{e}^{-ik{t}^{\prime }}\widehat{F}\left(k,t^{\prime }\right)dk\Vert }_{H_t^s\left(\mathbb{R}\right)}d{t}^{\prime }\hfill \\ \hfill & +{\int }_0^T{\Vert {\displaystyle \frac{1}{2\pi }}{\int }_0^{\infty }{e}^{ikt}{e}^{ikx}{e}^{-ik{t}^{\prime }}\widehat{F}\left(k,t^{\prime }\right)dk\Vert }_{H_t^s\left(\mathbb{R}\right)}d{t}^{\prime }\hfill \\ \hfill & +{\int }_0^T{\Vert {\displaystyle \frac{1}{2\pi }}{\int }_{-\infty }^0{e}^{ikt}{e}^{-ikx}{e}^{-ik{t}^{\prime }}\widehat{F}\left(-k,t^{\prime }\right)dk\Vert }_{H_t^s\left(\mathbb{R}\right)}d{t}^{\prime }\hfill \\ \hfill & \le {\int }_0^T{\left({\int }_{-\infty }^{\infty }{\left(1+k^2\right)}^s{\left|e^{-ikx}{e}^{-ik{t}^{\prime }}\widehat{F}\left(-k,t^{\prime }\right)\right|}^{2}dk\right)}^{{\displaystyle \frac{1}{2}}}d{t}^{\prime }\hfill \\ \hfill & +{\int }_0^T{\left({\int }_{-\infty }^{\infty }{\left(1+k^2\right)}^s{\left|e^{ikx}{e}^{-ik{t}^{\prime }}\widehat{F}\left(k,t^{\prime }\right)\right|}^{2}dk\right)}^{{\displaystyle \frac{1}{2}}}d{t}^{\prime }\hfill \\ \hfill & +{\int }_0^T{\left({\int }_{-\infty }^{\infty }{\left(1+k^2\right)}^s{\left|e^{ikx}{e}^{-ik{t}^{\prime }}\widehat{F}\left(k,t^{\prime }\right)\right|}^{2}dk\right)}^{{\displaystyle \frac{1}{2}}}d{t}^{\prime }\hfill \\ \hfill & +{\int }_0^T{\left({\int }_{-\infty }^{\infty }{\left(1+k^2\right)}^s{\left|e^{-ikx}{e}^{-ik{t}^{\prime }}\widehat{F}\left(-k,t^{\prime }\right)\right|}^{2}dk\right)}^{{\displaystyle \frac{1}{2}}}d{t}^{\prime },\hfill \\ \hfill & (\mathrm{by}\mathrm{Theorem}3\mathrm{and}\mathrm{Theorem}4)\hfill \\ \hfill & =4{\int }_0^T\Vert F\left(t^{\prime }\right){\Vert }_{H_x^s\left(\mathbb{R}\right)}d{t}^{\prime }\lesssim T\underset{t\in [0,T]}{sup}\Vert F\left(t\right){\Vert }_{H_x^s\left(\mathbb{R}\right)}\le \sqrt{T}\underset{t\in [0,T]}{sup}{\Vert F\left(t\right)\Vert }_{H_x^s\left(\mathbb{R}\right)}.\hfill \end{array}$$

Consequently, we obtain the following inequality:

$$\begin{array}{c}\hfill \Vert W_x\left(x,t\right){\Vert }_{H_t^s\left(\mathbb{R}\right)}\lesssim \sqrt{T}\underset{t\in [0,T]}{sup}{\Vert F\left(t\right)\Vert }_{H_x^s\left(\mathbb{R}\right)}.\end{array}$$

Since the inequality

$$\begin{array}{c}\hfill \Vert W_x\left(x,t\right){\Vert }_{H_t^{s-1}\left(\mathbb{R}\right)}\le {\Vert W_x\left(x,t\right)\Vert }_{H_t^s\left(\mathbb{R}\right)},\end{array}$$

we obtain the inequality

$$\begin{array}{c}\hfill \Vert W_x\left(x,t\right){\Vert }_{H_t^{s-1}\left(\mathbb{R}\right)}\lesssim \sqrt{T}\underset{t\in [0,T]}{sup}{\Vert F\left(t\right)\Vert }_{H_x^s\left(\mathbb{R}\right)}.\end{array}$$

As a result, we establish the inequality (67) as follows:

$$\begin{array}{c}\hfill \underset{x\in [0,\infty )}{sup}\Vert W_x\left(x,t\right){\Vert }_{H_t^{s-1}(0,T)}\lesssim \sqrt{T}\underset{t\in [0,T]}{sup}{\Vert F\left(t\right)\Vert }_{H_x^s\left(\mathbb{R}\right)}.\end{array}$$

We finish this proof of Lemma 4. □

Now, by applying Lemma 4 to Formulas (64) and (65), we derive the following inequalities:

$$\begin{array}{cc}\hfill \Vert G_0{\Vert }_{H_t^{s-1}(0,T)}& \le \Vert {\gamma }_1{\left(t\right)\Vert }_{H_t^{s-1}(0,T)}+\Vert U_x\left(0,t\right){\Vert }_{H_t^{s-1}(0,T)}+{\Vert W_x\left(0,t\right)\Vert }_{H_t^{s-1}(0,T)}\hfill \\ \hfill & \lesssim \Vert {\gamma }_1{\left(t\right)\Vert }_{H_t^{s-1}(0,T)}+\left(\Vert U_0{\Vert }_{H_x^s\left(\mathbb{R}\right)}+{\Vert U_{*}\Vert }_{H_x^s\left(\mathbb{R}\right)}\right)+\sqrt{T}\underset{t\in [0,T]}{sup}{\Vert F\left(t\right)\Vert }_{H_x^s\left(\mathbb{R}\right)}\hfill \\ \hfill & \left(\mathrm{by}\left(66\right)\mathrm{and}\left(67\right)\right)\hfill \\ \hfill & \lesssim \Vert u_0{\Vert }_{H_x^s(0,\infty )}+\Vert u_{*}{\Vert }_{H_x^s(0,\infty )}+\Vert {\gamma }_1{\Vert }_{H_t^{s-1}(0,T)}+\sqrt{T}\underset{t\in [0,T]}{sup}{\Vert f\left(t\right)\Vert }_{H_x^s(0,\infty )}\hfill \\ \hfill & \left(\mathrm{by}\left(26\right)–\left(28\right)\right),\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill \Vert H_0{\Vert }_{H_t^{s-1}(0,T)}& \le {\varpi }_1\Vert U\left(0,t\right){\Vert }_{H_t^{s-1}(0,T)}+{\varpi }_1{\Vert W\left(0,t\right)\Vert }_{H_t^{s-1}(0,T)}\hfill \\ \hfill & \le {\varpi }_1\Vert U\left(0,t\right){\Vert }_{H_t^s(0,T)}+{\varpi }_1{\Vert W\left(0,t\right)\Vert }_{H_t^s(0,T)}\hfill \\ \hfill & \le {\varpi }_1\left(\Vert U_0{\Vert }_{H_x^s\left(\mathbb{R}\right)}+{\Vert U_{*}\Vert }_{H_x^s\left(\mathbb{R}\right)}\right)+{\varpi }_1\left(\sqrt{T}\underset{t\in [0,T]}{sup}{\Vert F\left(t\right)\Vert }_{H_x^s\left(\mathbb{R}\right)}\right)\hfill \\ \hfill & \left(\mathrm{by}\left(42\right)\mathrm{and}\left(57\right)\right)\hfill \\ \hfill & \lesssim \Vert u_0{\Vert }_{H_x^s(0,\infty )}+\Vert u_{*}{\Vert }_{H_x^s(0,\infty )}+\sqrt{T}\underset{t\in [0,T]}{sup}{\Vert f\left(t\right)\Vert }_{H_x^s(0,\infty )}\hfill \\ \hfill & \left(\mathrm{by}\left(26\right)–\left(28\right)\right).\hfill \end{array}$$

According to Equation (39),

$$\begin{array}{c}\hfill S_1[u_0,u_{*},{\gamma }_1;f](x,t)=\widetilde{S_1}[U_0,U_{*};0]{{|}_{x>0}+\widetilde{S_1}[0,0;F]|}_{x>0}+S_1[0,0,G_0;0](x,t)+S_1[0,0,H_0;0](x,t),\end{array}$$

we obtain the following inequality:

$$\begin{array}{cc}\hfill & \underset{t\in [0,T]}{sup}{\Vert S_1[u_0,u_{*},{\gamma }_1;f](x,t)\Vert }_{H_x^s(0,\infty )}\hfill \\ \hfill & \le \underset{t\in [0,T]}{sup}\Vert \widetilde{S_1}[U_0,U_{*};0]\left(t\right){\Vert }_{H_x^s(0,\infty )}+\underset{t\in [0,T]}{sup}{\Vert \widetilde{S_1}[0,0;F]\left(t\right)\Vert }_{H_x^s(0,\infty )}\hfill \\ \hfill & +\underset{t\in [0,T]}{sup}\Vert S_1[0,0,G_0;0]\left(t\right){\Vert }_{H_x^s(0,\infty )}+\underset{t\in [0,T]}{sup}{\Vert S_1[0,0,H_0;0]\left(t\right)\Vert }_{H_x^s(0,\infty )}\hfill \\ \hfill & \lesssim \Vert U_0{\Vert }_{H_x^s\left(\mathbb{R}\right)}+\Vert U_{*}{\Vert }_{H_x^s\left(\mathbb{R}\right)}+T\underset{t\in [0,T]}{sup}\Vert F\left(t\right){\Vert }_{H_x^s\left(\mathbb{R}\right)}+\Vert G_0{\left(t\right)\Vert }_{H_t^{s-1}(0,T)}+{\Vert H_0\left(t\right)\Vert }_{H_t^{s-1}(0,T)}\hfill \\ \hfill & (\mathrm{by}\left(18\right),\left(22\right),\left(40\right)\mathrm{and}\left(55\right))\hfill \\ \hfill & \lesssim \Vert u_0{\Vert }_{H_x^s(0,\infty )}+\Vert u_{*}{\Vert }_{H_x^s(0,\infty )}+T\underset{t\in [0,T]}{sup}\Vert f\left(t\right){\Vert }_{H_x^s(0,\infty )}+{\Vert {\gamma }_1\Vert }_{H_t^{s-1}(0,T)}\hfill \\ \hfill & +\sqrt{T}\underset{t\in [0,T]}{sup}{\Vert f\left(t\right)\Vert }_{H_x^s(0,\infty )}(\mathrm{by}\left(26\right)–\left(28\right)\mathrm{and}\left(64\right)–\left(67\right))\hfill \\ \hfill & \lesssim \Vert u_0{\Vert }_{H_x^s(0,\infty )}+\Vert u_{*}{\Vert }_{H_x^s(0,\infty )}+\Vert {\gamma }_1{\Vert }_{H_t^{s-1}(0,T)}+\sqrt{T}\underset{t\in [0,T]}{sup}{\Vert f\left(t\right)\Vert }_{H_x^s(0,\infty )}.\hfill \end{array}$$

Consequently, we obtain the following inequality:

$$\begin{array}{c}\hfill \underset{t\in [0,T]}{sup}\Vert u\left(t\right){\Vert }_{H_x^s(0,\infty )}\lesssim \Vert u_0{\Vert }_{H_x^s(0,\infty )}+\Vert u_{*}{\Vert }_{H_x^s(0,\infty )}+\Vert {\gamma }_1{\Vert }_{H_t^{s-1}(0,T)}+\sqrt{T}\underset{t\in [0,T]}{sup}{\Vert f\left(t\right)\Vert }_{H_x^s(0,\infty )},\end{array}$$

(68)

and

$$\begin{array}{cc}\hfill & \underset{x\in [0,\infty )}{sup}{\Vert S_1[u_0,u_{*},{\gamma }_1;f](x,t)\Vert }_{H_t^s(0,T)}\hfill \\ \hfill & \le \underset{x\in [0,\infty )}{sup}\Vert \widetilde{S_1}[U_0,U_{*};0]\left(x\right){\Vert }_{H_t^s(0,T)}+\underset{x\in [0,\infty )}{sup}{\Vert \widetilde{S_1}[0,0;F]\left(x\right)\Vert }_{H_t^s(0,T)}\hfill \\ \hfill & +\underset{x\in [0,\infty )}{sup}\Vert S_1[0,0,G_0;0]\left(x\right){\Vert }_{H_t^s(0,T)}+\underset{x\in [0,\infty )}{sup}{\Vert S_1[0,0,H_0;0]\left(x\right)\Vert }_{H_t^s(0,T)}\hfill \\ \hfill & \lesssim \Vert U_0{\Vert }_{H_x^s\left(\mathbb{R}\right)}+\Vert U_{*}{\Vert }_{H_x^s\left(\mathbb{R}\right)}+\sqrt{T}\underset{t\in [0,T]}{sup}\Vert F\left(t\right){\Vert }_{H_x^s\left(\mathbb{R}\right)}+{\Vert G_0\left(t\right)\Vert }_{H_t^{s-1}(0,T)}\hfill \\ \hfill & +\Vert H_0{\left(t\right)\Vert }_{H_t^{s-1}(0,T)}(\mathrm{by}\left(18\right),\left(24\right),\left(42\right)\mathrm{and}\left(57\right))\hfill \\ \hfill & \lesssim \Vert u_0{\Vert }_{H_x^s(0,\infty )}+\Vert u_{*}{\Vert }_{H_x^s(0,\infty )}+\Vert {\gamma }_1{\Vert }_{H_t^{s-1}(0,T)}+\sqrt{T}\underset{t\in [0,T]}{sup}{\Vert f\left(t\right)\Vert }_{H_x^s(0,\infty )}\hfill \\ \hfill & (\mathrm{by}\left(26\right)–\left(28\right),\left(42\right),\left(57\right),\left(66\right)\mathrm{and}\left(67\right)).\hfill \end{array}$$

Therefore, we derive the following inequality:

$$\begin{array}{c}\hfill \underset{x\in [0,\infty )}{sup}\Vert u\left(x\right){\Vert }_{H_t^s(0,T)}\lesssim \Vert u_0{\Vert }_{H_x^s(0,\infty )}+\Vert u_{*}{\Vert }_{H_x^s(0,\infty )}+\Vert {\gamma }_1{\Vert }_{H_t^{s-1}(0,T)}+\sqrt{T}\underset{t\in [0,T]}{sup}{\Vert f\left(t\right)\Vert }_{H_x^s(0,\infty )}.\end{array}$$

(69)

We combine Equations (68) and (69) and establish the following inequality (13):

$$\begin{array}{cc}\hfill & \underset{t\in [0,T]}{sup}{\Vert u\left(t\right)\Vert }_{H_x^s(0,\infty )}+\underset{x\in [0,\infty )}{sup}{\Vert u\left(x\right)\Vert }_{H_t^s(0,T)}\hfill \\ \hfill & \le C_s\left(\Vert u_0{\Vert }_{H_x^s(0,\infty )}+\Vert u_{*}{\Vert }_{H_x^s(0,\infty )}+\Vert {\gamma }_1{\Vert }_{H_t^{s-1}(0,T)}+\sqrt{T}\underset{t\in [0,T]}{sup}{\Vert f\left(t\right)\Vert }_{H_x^s(0,\infty )}\right),\hfill \end{array}$$

where $C_s>0$ is a constant depending on s.

By employing a similar proof strategy as in (13), we can derive the inequality (14) below:

$$\begin{array}{cc}\hfill & \underset{t\in [0,T]}{sup}{\Vert v\left(t\right)\Vert }_{H_x^s(0,\infty )}+\underset{x\in [0,\infty )}{sup}{\Vert v\left(x\right)\Vert }_{H_t^s(0,T)}\hfill \\ \hfill & \le d_s\left(\Vert v_0{\Vert }_{H_x^s(0,\infty )}+\Vert v_{*}{\Vert }_{H_x^s(0,\infty )}+\Vert {\gamma }_2{\Vert }_{H_t^{s-1}(0,T)}+\sqrt{T}\underset{t\in [0,T]}{sup}{\Vert g\left(t\right)\Vert }_{H_x^s(0,\infty )}\right),\hfill \end{array}$$

where $d_s>0$ is a constant depending on s. We finish this proof of Theorem 1.

Now, by applying Theorem 1 and combining the results from Equations (13) and (14), we obtain that

$$\begin{array}{cc}\hfill & \underset{t\in [0,T]}{sup}{\Vert u\left(t\right)\Vert }_{H_x^s(0,\infty )}+\underset{x\in [0,\infty )}{sup}{\Vert u\left(x\right)\Vert }_{H_t^s(0,T)}+\underset{t\in [0,T]}{sup}{\Vert v\left(t\right)\Vert }_{H_x^s(0,\infty )}+\underset{x\in [0,\infty )}{sup}{\Vert v\left(x\right)\Vert }_{H_t^s(0,T)}\hfill \\ \hfill & \le C_s^{*}\left(\Vert u_0{\Vert }_{H_x^s(0,\infty )}+\Vert u_{*}{\Vert }_{H_x^s(0,\infty )}+\Vert v_0{\Vert }_{H_x^s(0,\infty )}+\Vert v_{*}{\Vert }_{H_x^s(0,\infty )}+{\Vert {\gamma }_1\Vert }_{H_t^{s-1}(0,T)}\right.\hfill \\ \hfill & +\left.\Vert {\gamma }_2{\Vert }_{H_t^{s-1}(0,T)}+\sqrt{T}\underset{t\in [0,T]}{sup}{\Vert f\left(t\right)\Vert }_{H_x^s(0,\infty )}+\sqrt{T}\underset{t\in [0,T]}{sup}{\Vert g\left(t\right)\Vert }_{H_x^s(0,\infty )}\right),\hfill \end{array}$$

(70)

where $C_s^{*}=max\{C_s,d_s\}$.

5. The Proof of Therorem 2 (about the Local Well-Posedness of the Coupled System of Wave Equations in Sobolev Spaces)

In this section, we begin by introducing the iteration map. Following this, Lemma 5 and Lemma 6 are presented to demonstrate that the iteration map is both a contraction and a self-map on a closed ball. By applying the contraction mapping theorem, we prove the uniqueness of the solution. Additionally, Lemma 7 establishes that the data-to-solution map is locally Lipschitz continuous. By leveraging these results, we conclude the proof of Theorem 2.

5.1. Existence and Uniqueness

In this subsection, we prove the existence and uniqueness of the solution for (1).

First, for $0<T<1$ and some $T^{*}\in (0,T)$, we set $f=u^{2}v$ and $g=u{v}^2$ in (9) and (11), respectively. And then, for data $D_1=\left(u_0,u_{*},v_0,v_{*},{\gamma }_1,{\gamma }_2\right)$, we define the iteration map $\left(u,v\right)⟼{\mathcal{F}}_{D_1}\left(u,v\right)\doteq \left({\Phi }_{T^{*}}\left(u,v\right),{\Psi }_{T^{*}}\left(u,v\right)\right)$, which is derived from Formulas (9) and (11) for the forced-linear wave-equation IBVPs (6) and (7). More precisely, we have

$$\begin{array}{c}\hfill {\Phi }_{T^{*}}\left(u,v\right)\doteq S_1[u_0,u_{*},{\gamma }_1;u^{2}v],{\Psi }_{T^{*}}\left(u,v\right)\doteq S_1[v_0,v_{*},{\gamma }_2;u{v}^2].\end{array}$$

We will demonstrate that the iteration map

$$\begin{array}{c}\hfill {\mathcal{F}}_{D_1}\left(u,v\right)\doteq \left({\Phi }_{T^{*}}\left(u,v\right),{\Psi }_{T^{*}}\left(u,v\right)\right)=\left(S_1[u_0,u_{*},{\gamma }_1;u^{2}v],S_1[v_0,v_{*},{\gamma }_2;u{v}^2]\right)\end{array}$$

is a contraction in the Banach space

$$\begin{array}{c}\hfill X={\left(C\left([0,T^{*}];H_x^s\left(0,\infty \right)\right)\cap C\left([0,\infty );H_t^s\left(0,T^{*}\right)\right)\right)}^2,\end{array}$$

(71)

with the norm

$$\begin{array}{cc}\hfill \Vert \left(u,v\right){\Vert }_X& =\underset{t\in [0,T^{*}]}{sup}{\Vert u\left(t\right)\Vert }_{H_x^s(0,\infty )}+\underset{x\in [0,\infty )}{sup}{\Vert u\left(x\right)\Vert }_{H_t^s(0,T^{*})}\hfill \\ \hfill & +\underset{t\in [0,T^{*}]}{sup}{\Vert v\left(t\right)\Vert }_{H_x^s(0,\infty )}+\underset{x\in [0,\infty )}{sup}{\Vert v\left(x\right)\Vert }_{H_t^s(0,T^{*})}.\hfill \end{array}$$

Next, we begin to prove that the map $\left(u,v\right)⟼{\mathcal{F}}_{D_1}\left(u,v\right)$ is onto X. We consider a closed ball $B\left(0,r\right)=\{\left(u,v\right)\in X:\Vert \left(u,v\right){\Vert }_X\le r\}$, where

$$\begin{array}{c}\hfill C_s^{*}=max\{C_s,d_s\},r\doteq 2C_s^{*}{\Vert \left(u_0,u_{*},v_0,v_{*},{\gamma }_1,{\gamma }_2\right)\Vert }_D\end{array}$$

and

$$\begin{array}{cc}\hfill \Vert \left(u_0,u_{*},v_0,v_{*},{\gamma }_1,{\gamma }_2\right){\Vert }_D& =\Vert u_0{\Vert }_{H_x^s(0,\infty )}+\Vert u_{*}{\Vert }_{H_x^s(0,\infty )}+{\Vert v_0\Vert }_{H_x^s(0,\infty )}\hfill \\ \hfill & +\Vert v_{*}{\Vert }_{H_x^s(0,\infty )}+\Vert {\gamma }_1{\Vert }_{H_t^{s-1}(0,T)}+{\Vert {\gamma }_2\Vert }_{H_t^{s-1}(0,T)}.\hfill \end{array}$$

(72)

In the next lemma, we determine the condition on $T^{*}$ under which ${\mathcal{F}}_{D_1}$ maps onto $B(0,r)$.

Lemma 5.

Let $C_s^{*}=max\{C_s,d_s\}$ and $r=2C_s^{*}{\Vert \left(u_0,u_{*},v_0,v_{*},{\gamma }_1,{\gamma }_2\right)\Vert }_D$. If the following condition for $T^{*}$ given by

$$\begin{array}{c}\hfill 0<T^{*}\le min\left\{T,{\displaystyle \frac{1}{256{\left(C_s^{*}\right)}^{10}{\Vert \left(u_0,u_{*},v_0,v_{*},{\gamma }_1,{\gamma }_2\right)\Vert }_D^4}}\right\}\end{array}$$

(73)

is satisfied, then the iteration map ${\mathcal{F}}_{D_1}$ is onto $B(0,r)$.

Proof. 

For $(u,v)\in B(0,r)$, the following inequality holds:

$$\begin{array}{cc}\hfill & \Vert {\mathcal{F}}_{D_1}{(u,v)\Vert }_X={\Vert \left(S_1[u_0,u_{*},{\gamma }_1;u^{2}v],S_1[v_0,v_{*},{\gamma }_2;u{v}^2]\right)\Vert }_X\hfill \\ \hfill & =\underset{t\in [0,T^{*}]}{sup}\Vert S_1[u_0,u_{*},{\gamma }_1;u^{2}v]\left(t\right){\Vert }_{H_x^s(0,\infty )}+\underset{x\in [0,\infty )}{sup}{\Vert S_1[u_0,u_{*},{\gamma }_1;u^{2}v]\left(x\right)\Vert }_{H_t^s(0,T^{*})}\hfill \\ \hfill & +\underset{t\in [0,T^{*}]}{sup}\Vert S_1[v_0,v_{*},{\gamma }_2;u{v}^2]\left(t\right){\Vert }_{H_x^s(0,\infty )}+\underset{x\in [0,\infty )}{sup}{\Vert S_1[v_0,v_{*},{\gamma }_2;u{v}^2]\left(x\right)\Vert }_{H_t^s(0,T^{*})}\hfill \\ \hfill & \le C_s^{*}\left(\Vert u_0{\Vert }_{H_x^s(0,\infty )}+\Vert u_{*}{\Vert }_{H_x^s(0,\infty )}+\Vert v_0{\Vert }_{H_x^s(0,\infty )}+\Vert v_{*}{\Vert }_{H_x^s(0,\infty )}+{\Vert {\gamma }_1\Vert }_{H_t^{s-1}(0,T)}\right.\hfill \\ \hfill & +\Vert {\gamma }_2{\Vert }_{H_t^{s-1}(0,T)}+\left.\sqrt{T^{*}}\underset{t\in [0,T^{*}]}{sup}\Vert u^2{v\Vert }_{H_x^s(0,\infty )}+\sqrt{T^{*}}\underset{t\in [0,T^{*}]}{sup}{\Vert u{v}^2\Vert }_{H_x^s(0,\infty )}\right),\left(\mathrm{by}\left(70\right)\right)\hfill \\ \hfill & ={\displaystyle \frac{r}{2}}+C_s^{*}\sqrt{T^{*}}\left(\underset{t\in [0,T^{*}]}{sup}\Vert u^2{v\Vert }_{H_x^s(0,\infty )}+\underset{t\in [0,T^{*}]}{sup}{\Vert u{v}^2\Vert }_{H_x^s(0,\infty )}\right)\hfill \\ \hfill & \le {\displaystyle \frac{r}{2}}+C_s^{*}\sqrt{T^{*}}\left(C_s^2{r}^3+C_s^2{r}^3\right),\left(\mathrm{by}\mathrm{Lemma}2\right)\hfill \\ \hfill & \le {\displaystyle \frac{r}{2}}+2{\left(C_s^{*}\right)}^3\sqrt{T^{*}}{r}^3.\hfill \end{array}$$

Hence, we obtain the inequality

$$\begin{array}{c}\hfill \Vert {\mathcal{F}}_{D_1}{(u,v)\Vert }_X\le {\displaystyle \frac{r}{2}}+2{\left(C_s^{*}\right)}^3\sqrt{T^{*}}{r}^3.\end{array}$$

For $0<T^{*}\le T$, in order to prove that ${\mathcal{F}}_{D_1}$ is onto on $B(0,r)$, we aim for the following inequality to hold:

$$\begin{array}{c}\hfill {\displaystyle \frac{r}{2}}+2{\left(C_s^{*}\right)}^3\sqrt{T^{*}}{r}^3\le r,\end{array}$$

which is equivalent to

$$\begin{array}{c}\hfill 0<T^{*}\le {\displaystyle \frac{1}{256{\left(C_s^{*}\right)}^{10}{\Vert \left(u_0,u_{*},v_0,v_{*},{\gamma }_1,{\gamma }_2\right)\Vert }_D^4}}.\end{array}$$

Thus, when $T^{*}$ satisfies the following condition (73):

$$\begin{array}{c}\hfill 0<T^{*}\le min\left\{T,{\displaystyle \frac{1}{256{\left(C_s^{*}\right)}^{10}{\Vert \left(u_0,u_{*},v_0,v_{*},{\gamma }_1,{\gamma }_2\right)\Vert }_D^4}}\right\},\end{array}$$

it follows that ${\mathcal{F}}_{D_1}$ is onto $B(0,r)$. □

Now, we begin to prove that the map $\left(u,v\right)⟼{\mathcal{F}}_{D_1}\left(u,v\right)$ is a contraction in X. We establish the constraint on $T^{*}$ such that ${\mathcal{F}}_{D_1}$ is a contraction on $B(0,r)$, for $C_s^{*}=max\{C_s,d_s\}$ and $r=2C_s^{*}{\Vert \left(u_0,u_{*},v_0,v_{*},{\gamma }_1,{\gamma }_2\right)\Vert }_D$. This is detailed in the following lemma.

Lemma 6.

Let $C_s^{*}=max\{C_s,d_s\}$ and $r=2C_s^{*}{\Vert \left(u_0,u_{*},v_0,v_{*},{\gamma }_1,{\gamma }_2\right)\Vert }_D$. If the following condition for $T^{*}$ given by

$$\begin{array}{c}\hfill 0<T^{*}\le min\left\{T,{\displaystyle \frac{1}{2304{\left(C_s^{*}\right)}^{10}{\Vert \left(u_0,u_{*},v_0,v_{*},{\gamma }_1,{\gamma }_2\right)\Vert }_D^4}}\right\}\end{array}$$

(74)

is satisfied, then the iteration map ${\mathcal{F}}_{D_1}$ is a contraction on $B(0,r)$.

Proof. 

For $({\overline{u}}_1,{\overline{v}}_1),({\overline{u}}_2,{\overline{v}}_2)\in B(0,r)$, the following inequality holds:

$$\begin{array}{cc}\hfill & \Vert {\mathcal{F}}_{D_1}({\overline{u}}_1,{\overline{v}}_1)-{\mathcal{F}}_{D_1}({\overline{u}}_2,{\overline{v}}_2){\Vert }_X\hfill \\ \hfill & =\Vert \left(S_1[u_0,u_{*},{\gamma }_1;{\overline{u}}_1^2{\overline{v}}_1],S_1[v_0,v_{*},{\gamma }_2;{\overline{u}}_1{\overline{v}}_1^2]\right)-\left(S_1[u_0,u_{*},{\gamma }_1;{\overline{u}}_2^2{\overline{v}}_2],S_1[v_0,v_{*},{\gamma }_2;{\overline{u}}_2{\overline{v}}_2^2]\right){\Vert }_X\hfill \\ \hfill & =\Vert \left(S_1[0,0,0;{\overline{u}}_1^2{\overline{v}}_1-{\overline{u}}_2^2{\overline{v}}_2],S_1[0,0,0;{\overline{u}}_1{\overline{v}}_1^2-{\overline{u}}_2{\overline{v}}_2^2]\right){\Vert }_X\hfill \\ \hfill & =\underset{t\in [0,T^{*}]}{sup}{\Vert S_1[0,0,0;{\overline{u}}_1^2{\overline{v}}_1-{\overline{u}}_2^2{\overline{v}}_2]\left(t\right)\Vert }_{H_x^s(0,\infty )}\hfill \\ \hfill & +\underset{x\in [0,\infty )}{sup}{\Vert S_1[0,0,0;{\overline{u}}_1^2{\overline{v}}_1-{\overline{u}}_2^2{\overline{v}}_2]\left(x\right)\Vert }_{H_t^s(0,T^{*})}\hfill \\ \hfill & +\underset{t\in [0,T^{*}]}{sup}{\Vert S_1[0,0,0;{\overline{u}}_1{\overline{v}}_1^2-{\overline{u}}_2{\overline{v}}_2^2]\left(t\right)\Vert }_{H_x^s(0,\infty )}\hfill \\ \hfill & +\underset{x\in [0,\infty )}{sup}{\Vert S_1[0,0,0;{\overline{u}}_1{\overline{v}}_1^2-{\overline{u}}_2{\overline{v}}_2^2]\left(x\right)\Vert }_{H_t^s(0,T^{*})}\hfill \\ \hfill & \le C_s^{*}\left(\sqrt{T^{*}}\underset{t\in [0,T^{*}]}{sup}\Vert {\overline{u}}_1^2{\overline{v}}_1-{\overline{u}}_2^2{\overline{v}}_2{\Vert }_{H_x^s(0,\infty )}+\sqrt{T^{*}}\underset{t\in [0,T^{*}]}{sup}{\Vert {\overline{u}}_1{\overline{v}}_1^2-{\overline{u}}_2{\overline{v}}_2^2\Vert }_{H_x^s(0,\infty )}\right),\hfill \\ \hfill & (\mathrm{by}\left(70\right))\hfill \\ \hfill & \le C_s^{*}\sqrt{T^{*}}\left(\underset{t\in [0,T^{*}]}{sup}\Vert {\overline{u}}_1^2({\overline{v}}_1-{\overline{v}}_2){\Vert }_{H_x^s(0,\infty )}+\underset{t\in [0,T^{*}]}{sup}{\Vert ({\overline{u}}_1+{\overline{u}}_2)({\overline{u}}_1-{\overline{u}}_2){\overline{v}}_2\Vert }_{H_x^s(0,\infty )}\right.\hfill \\ \hfill & \left.+\underset{t\in [0,T^{*}]}{sup}\Vert {\overline{v}}_1^2({\overline{u}}_1-{\overline{u}}_2){\Vert }_{H_x^s(0,\infty )}+\underset{t\in [0,T^{*}]}{sup}{\Vert ({\overline{v}}_1-{\overline{v}}_2)({\overline{v}}_1+{\overline{v}}_2){\overline{u}}_2\Vert }_{H_x^s(0,\infty )}\right)\hfill \\ \hfill & \le 6{\left(C_s^{*}\right)}^3{r}^2\sqrt{T^{*}}{\Vert ({\overline{u}}_1,{\overline{v}}_1)-({\overline{u}}_2,{\overline{v}}_2)\Vert }_X\hfill \end{array}$$

For $0<T^{*}\le T$, in order to prove that ${\mathcal{F}}_{D_1}$ is a contraction, we aim for the following inequality to hold:

$$6{\left(C_s^{*}\right)}^3{r}^2\sqrt{T^{*}}\le {\displaystyle \frac{1}{2}}$$

which is equivalent to

$$\begin{array}{c}\hfill 0<T^{*}\le min{\displaystyle \frac{1}{2304{\left(C_s^{*}\right)}^{10}{\Vert \left(u_0,u_{*},v_0,v_{*},{\gamma }_1,{\gamma }_2\right)\Vert }_D^4}}.\end{array}$$

Thus, when $T^{*}$ satisfies the following condition (74):

$$\begin{array}{c}\hfill 0<T^{*}\le min\left\{T,{\displaystyle \frac{1}{2304{\left(C_s^{*}\right)}^{10}{\Vert \left(u_0,u_{*},v_0,v_{*},{\gamma }_1,{\gamma }_2\right)\Vert }_D^4}}\right\}\end{array}$$

it follows that ${\mathcal{F}}_{D_1}$ is a contraction. □

We will now set the lifespan as follows:

$$\begin{array}{c}\hfill T^{*}=min\left\{T,{\displaystyle \frac{1}{2304{\left(C_s^{*}\right)}^{10}{\Vert \left(u_0,u_{*},v_0,v_{*},{\gamma }_1,{\gamma }_2\right)\Vert }_D^4}}\right\}.\end{array}$$

(75)

This choice ensures that $T^{*}$ satisfies both conditions (73) and (74). As a result, the iteration map ${\mathcal{F}}_{D_1}$ acts as a contraction and maps onto $B(0,r)$. Consequently, by applying the contraction mapping theorem, we conclude that Equation $(u,v)={\mathcal{F}}_{D_1}(u,v)$ has a unique solution $(u,v)\in B(0,r)\subset X.$

5.2. Continuity of the Data-to-Solution Map

In this subsection, we will establish that the data-to-solution map $(u_0,u_{*},v_0,v_{*},r_1,r_2)$$⟼(u,v)$ is locally Lipschitz continuous. This step is crucial for completing the proof of local well-posedness for the IBVP (1) on the half-line.

We consider two distinct sets of data $D_1=(u_0,u_{*},v_0,v_{*},r_1,r_2)$ and $D_2=({\mathcal{U}}_0,{\mathcal{U}}_{*},{\mathcal{V}}_0,$${\mathcal{V}}_{*},{\mathcal{R}}_1,{\mathcal{R}}_2)$. These datasets reside within a ball $B_{\rho }\subset D$ of radius $\rho >0$ centered at a distance $\mathcal{R}$ from the origin, where

$$D=H_x^s(0,\infty )\times H_x^s(0,\infty )\times H_x^s(0,\infty )\times H_x^s(0,\infty )\times H_t^{s-1}(0,T)\times H_t^{s-1}(0,T)$$

with the norm (72).

Let $(u,v)={\mathcal{F}}_{D_1}(u,v)$ and $(\mathcal{U},\mathcal{V})={\mathcal{F}}_{D_2}(\mathcal{U},\mathcal{V})$ represent the solutions corresponding to the IBVP (1). The lifespans of $(u,v)$ and $(\mathcal{U},\mathcal{V})$ are denoted by $T_{(u,v)}$ and $T_{(\mathcal{U},\mathcal{V})}$, respectively, where

$$\begin{array}{cc}\hfill T_{(u,v)}=& min\left\{T,{\displaystyle \frac{1}{2304{\left(C_s^{*}\right)}^{10}{\Vert \left(u_0,u_{*},v_0,v_{*},{\gamma }_1,{\gamma }_2\right)\Vert }_D^4}}\right\},\hfill \\ \hfill T_{(\mathcal{U},\mathcal{V})}=& min\left\{T,{\displaystyle \frac{1}{2304{\left(C_s^{*}\right)}^{10}{\Vert \left({\mathcal{U}}_0,{\mathcal{U}}_{*},{\mathcal{V}}_0,{\mathcal{V}}_{*},{\mathcal{R}}_1,{\mathcal{R}}_2\right)\Vert }_D^4}}\right\}.\hfill \end{array}$$

Given that

$$max\left\{\Vert \left(u_0,u_{*},v_0,v_{*},{\gamma }_1,{\gamma }_2\right){\Vert }_D,{\Vert \left({\mathcal{U}}_0,{\mathcal{U}}_{*},{\mathcal{V}}_0,{\mathcal{V}}_{*},{\mathcal{R}}_1,{\mathcal{R}}_2\right)\Vert }_D\right\}\le \rho +\mathcal{R},$$

we can deduce that

$$min\left\{T_{(u,v)},T_{(\mathcal{U},\mathcal{V})}\right\}\ge min\left\{T,{\displaystyle \frac{1}{2304{\left(C_s^{*}\right)}^{10}{\left(\mathcal{R}+\rho \right)}^4}}\right\}\doteq T_{\eta }.$$

Thus, both solutions $(u,v)$ and $(\mathcal{U},\mathcal{V})$ are guaranteed to exist for any $0<t\le T_{\eta }$. For the lifespan $T_{\eta }$, we define the solution space $X_{\eta }$ as the solution space X outlined in (71) with $T^{*}=T_{\eta }$.

In the upcoming lemma, we will demonstrate that the data-to-solution map $(u_0,u_{*},v_0,$$v_{*},r_1,r_2)⟼(u,v)$ is locally Lipschitz continuous.

Lemma 7.

Let $C_s^{*}=max\{C_s,d_s\}$ and $r_{\eta }={\left(12{\left(C_s^{*}\right)}^3{T}_{\eta }^{{\displaystyle \frac{1}{2}}}\right)}^{-{\displaystyle \frac{1}{2}}}.$ For any $(u,v),(\mathcal{U},\mathcal{V})\in B(0,r_{\eta })\subset X_{\eta }$ with data in the ball $B_{\rho }$, we can establish the following inequality:

$$\begin{array}{c}\hfill {\Vert (u,v)-(\mathcal{U},\mathcal{V})\Vert }_{X_{\eta }}\le 2C_s^{*}{\Vert (u_0,u_{*},v_0,v_{*},r_1,r_2)-({\mathcal{U}}_0,{\mathcal{U}}_{*},{\mathcal{V}}_0,{\mathcal{V}}_{*},{\mathcal{R}}_1,{\mathcal{R}}_2)\Vert }_D.\end{array}$$

(76)

Hence, the data-to-solution map $(u_0,u_{*},v_0,v_{*},r_1,r_2)⟼(u,v)$ is locally Lipschitz continuous.

Proof. 

For any $(u,v),(\mathcal{U},\mathcal{V})\in B(0,r_{\eta })\subset X_{\eta }$ with data in the ball $B_{\rho }$, we derive the following inequality:

$$\begin{array}{cc}\hfill & {\Vert (u,v)-(\mathcal{U},\mathcal{V})\Vert }_{X_{\eta }}={\Vert {\mathcal{F}}_{D_1}(u,v)-{\mathcal{F}}_{D_2}(\mathcal{U},\mathcal{V})\Vert }_{X_{\eta }}\hfill \\ \hfill & =\Vert (S_1[u_0-{\mathcal{U}}_0,u_{*}-{\mathcal{U}}_{*},r_1-{\mathcal{R}}_1;u^{2}v-{\mathcal{U}}^2\mathcal{V}],S_1[v_0-{\mathcal{V}}_0,v_{*}-{\mathcal{V}}_{*},r_2-{\mathcal{R}}_2;u{v}^2-\mathcal{U}{\mathcal{V}}^2]){\Vert }_{X_{\eta }}\hfill \\ \hfill & =\underset{t\in [0,T_{\eta }]}{sup}{\Vert S_1[u_0-{\mathcal{U}}_0,u_{*}-{\mathcal{U}}_{*},r_1-{\mathcal{R}}_1;u^{2}v-{\mathcal{U}}^2\mathcal{V}]\left(t\right)\Vert }_{H_x^s(0,\infty )}\hfill \\ \hfill & +\underset{x\in [0,\infty )}{sup}{\Vert S_1[u_0-{\mathcal{U}}_0,u_{*}-{\mathcal{U}}_{*},r_1-{\mathcal{R}}_1;u^{2}v-{\mathcal{U}}^2\mathcal{V}]\left(x\right)\Vert }_{H_t^s(0,T_{\eta })}\hfill \\ \hfill & +\underset{t\in [0,T_{\eta }]}{sup}{\Vert S_1[v_0-{\mathcal{V}}_0,v_{*}-{\mathcal{V}}_{*},r_2-{\mathcal{R}}_2;u{v}^2-\mathcal{U}{\mathcal{V}}^2]\left(t\right)\Vert }_{H_x^s(0,\infty )}\hfill \\ \hfill & +\underset{x\in [0,\infty )}{sup}{\Vert S_1[v_0-{\mathcal{V}}_0,v_{*}-{\mathcal{V}}_{*},r_2-{\mathcal{R}}_2;u{v}^2-\mathcal{U}{\mathcal{V}}^2]\left(x\right)\Vert }_{H_t^s(0,T_{\eta })}\hfill \\ \hfill & \le C_s^{*}(\Vert u_0-{\mathcal{U}}_0{\Vert }_{H_x^s(0,\infty )}+\Vert u_{*}-{\mathcal{U}}_{*}{\Vert }_{H_x^s(0,\infty )}+\Vert v_0-{\mathcal{V}}_0{\Vert }_{H_x^s(0,\infty )}+{\Vert v_{*}-{\mathcal{V}}_{*}\Vert }_{H_x^s(0,\infty )}\hfill \\ \hfill & \Vert r_1-{\mathcal{R}}_1{\Vert }_{H_t^{s-1}(0,\infty )}+\Vert r_2-{\mathcal{R}}_2{\Vert }_{H_t^{s-1}(0,\infty )}+\sqrt{T_{\eta }}\underset{t\in [0,T_{\eta }]}{sup}{\Vert u^{2}v-{\mathcal{U}}^2\mathcal{V}\Vert }_{H_x^s(0,\infty )}\hfill \\ \hfill & +\sqrt{T_{\eta }}\underset{t\in [0,T_{\eta }]}{sup}\Vert u{v}^2-\mathcal{U}{\mathcal{V}}^2{\Vert }_{H_x^s(0,\infty )})(\mathrm{by}\left(70\right))\hfill \\ \hfill & \le C_s^{*}{\Vert (u_0,u_{*},v_0,v_{*},r_1,r_2)-({\mathcal{U}}_0,{\mathcal{U}}_{*},{\mathcal{V}}_0,{\mathcal{V}}_{*},{\mathcal{R}}_1,{\mathcal{R}}_2)\Vert }_D\hfill \\ \hfill & +C_s^{*}\sqrt{T_{\eta }}\left(\underset{t\in [0,T_{\eta }]}{sup}{\Vert v(u-\mathcal{U})(u+\mathcal{U})\Vert }_{H_x^s(0,\infty )}+\underset{t\in [0,T_{\eta }]}{sup}{\Vert (u-\mathcal{U})v^2\Vert }_{H_x^s(0,\infty )}\right.\hfill \\ \hfill & \left.+\underset{t\in [0,T_{\eta }]}{sup}{\Vert \mathcal{U}(v-\mathcal{V})(v+\mathcal{V})\Vert }_{H_x^s(0,\infty )}+\underset{t\in [0,T_{\eta }]}{sup}{\Vert {\mathcal{U}}^2(v-\mathcal{V})\Vert }_{H_x^s(0,\infty )}\right)\hfill \\ \hfill & \le C_s^{*}\Vert (u_0,u_{*},v_0,v_{*},r_1,r_2)-({\mathcal{U}}_0,{\mathcal{U}}_{*},{\mathcal{V}}_0,{\mathcal{V}}_{*},{\mathcal{R}}_1,{\mathcal{R}}_2){\Vert }_D+6C_s^{*}{C}_s^2{r}_{\eta }^2\sqrt{T_{\eta }}{\Vert (u,v)-(\mathcal{U},\mathcal{V})\Vert }_{X_{\eta }}\hfill \\ \hfill & \le C_s^{*}\Vert (u_0,u_{*},v_0,v_{*},r_1,r_2)-({\mathcal{U}}_0,{\mathcal{U}}_{*},{\mathcal{V}}_0,{\mathcal{V}}_{*},{\mathcal{R}}_1,{\mathcal{R}}_2){\Vert }_D+6{\left(C_s^{*}\right)}^3{r}_{\eta }^2\sqrt{T_{\eta }}{\Vert (u,v)-(\mathcal{U},\mathcal{V})\Vert }_{X_{\eta }}.\hfill \end{array}$$

Therefore, we derive the following inequality:

$$\begin{array}{c}\hfill {\Vert (u,v)-(\mathcal{U},\mathcal{V})\Vert }_{X_{\eta }}\le {\displaystyle \frac{C_s^{*}}{1-6{\left(C_s^{*}\right)}^3\sqrt{T_{\eta }}{r}_{\eta }^2}}{\Vert (u_0,u_{*},v_0,v_{*},r_1,r_2)-({\mathcal{U}}_0,{\mathcal{U}}_{*},{\mathcal{V}}_0,{\mathcal{V}}_{*},{\mathcal{R}}_1,{\mathcal{R}}_2)\Vert }_D.\end{array}$$

Hence, when we set

$$\begin{array}{c}\hfill r_{\eta }={\left(12{\left(C_s^{*}\right)}^3{T}_{\eta }^{{\displaystyle \frac{1}{2}}}\right)}^{-{\displaystyle \frac{1}{2}}},\end{array}$$

the following two inequalities hold:

$${\displaystyle \frac{1}{1-6{\left(C_s^{*}\right)}^3\sqrt{T_{\eta }}{r}_{\eta }^2}}\le 2\mathrm{and}1-6{\left(C_s^{*}\right)}^3\sqrt{T_{\eta }}{r}_{\eta }^2>0.$$

Consequently, we can derive (76), demonstrating that the data-to-solution map is locally Lipschitz continuous. This concludes the proof of Lemma 7. □

Now, we can proceed to prove Theorem 2. We define the lifespan as

$$\begin{array}{c}\hfill T^{*}=min\left\{T,{\displaystyle \frac{1}{2304{\left(C_s^{*}\right)}^{10}{\Vert (u_0,u_{*},v_0,v_{*},{\gamma }_1,{\gamma }_2)\Vert }_D^4}}\right\}.\end{array}$$

By utilizing Lemmas 5–7, we can finalize the proof of Theorem 2.

In conclusion, based on our previous experience, using classical methods to prove the local well-posedness of a system tends to be more cumbersome and restrictive compared to the UTM. The UTM facilitates the selection of appropriate Sobolev spaces for the boundary conditions of system (1), which simplifies subsequent estimates and the process of proving the local well-posedness of system (1). This article thus enhances our understanding of the utility of the UTM in establishing the local well-posedness of coupled wave equations, laying the foundation for future research on system (2).