Results from a Nonlinear Wave Equation with Acoustic and Fractional Boundary Conditions Coupling by Logarithmic Source and Delay Terms: Global Existence and Asymptotic Behavior of Solutions

Abstract

The nonlinear wave equation with acoustic and fractional boundary conditions, coupled with logarithmic source and delay terms, is significant for its ability to model complex systems, its contribution to the advancement of mathematical theory, and its wide-ranging applicability to real-world problems. This paper examines the global existence and general decay of solutions to a wave equation characterized by coupling with logarithmic source and delay terms, and governed by both fractional and acoustic boundary conditions. The global existence of solutions is analyzed under a range of hypotheses, and the general decay behavior is established through the construction and application of an appropriate Lyapunov function.

1. Introduction

The investigation of nonlinear wave equations has been fundamental in the analysis of diverse physical and engineering systems, encompassing areas such as fluid dynamics, elastic materials, and quantum field theory [1,2,3]. These equations typically govern the propagation of waves influenced by nonlinear forces, which can induce complex behaviors, including shock formation, dispersion, and wave breaking [4,5].

Significant research has been directed towards wave equations governed by acoustic boundary conditions and fractional derivatives within this context. Acoustic boundary conditions model the interaction between a wave-propagating medium and a solid boundary, leading to sound reflection and transmission phenomena. The study of nonlinear wave equations with acoustic and fractional boundary conditions, coupled with logarithmic source and delay terms, holds significant importance in both theoretical and applied mathematics. This research contributes to the mathematical theory by providing rigorous results on the global existence and long-term behavior of solutions, thus deepening our understanding of nonlinear wave equations under these intricate conditions.

We are interested in the following problem:

$$\left\{\begin{array}{cc}{u}_{tt}-\Delta u+{\mathfrak{A}}_1{u}_t+{\mathfrak{A}}_2{u}_t(x,t-\tau )={\left|u\right|}^{p-2}uln\left|u\right|,\hfill & x\in \Omega ,t>0,\hfill \\ {\displaystyle \frac{\partial u}{\partial \nu }}=-\mathfrak{B}{\partial }_t^{\alpha ,\eta }u+{\chi }_t,\hfill & x\in {\Sigma }_0,t>0,\hfill \\ u_t+\mathfrak{P}\left(x\right){\chi }_t+\mathfrak{Q}\left(x\right)\chi =0,\hfill & x\in {\Sigma }_0,t>0,\hfill \\ u(x,t)=0,\hfill & x\in {\Sigma }_1,t>0,\hfill \\ u(x,0)=u_0\left(x\right),u_t(x,0)=u_1\left(x\right),\hfill & x\in \Omega ,\hfill \\ u_t(x,t-\tau )=f_0(x,t-\tau ),\hfill & x\in \Omega ,t\in (0,\tau ),\hfill \\ \chi (x,0)={\chi }_0,\hfill & x\in {\Sigma }_0,\hfill \end{array}\right.$$

(1)

where $\nu$ is the outward normal to $\partial \Omega$, where ${\Sigma }_0$ and ${\Sigma }_1$ are closed and disjoint, and $\Omega$ is a bounded domain in ${\mathbb{R}}^n$, $n\ge 1$ with a suitably smooth border $\partial \Omega ={\Sigma }_0\cup {\Sigma }_1$ of class $C^2$. ${\mathfrak{A}}_1,\mathfrak{B}>0$, ${\mathfrak{A}}_2\in \mathbb{R}$, the coefficient $\tau >0$ expresses the time delay, $p>2$. The generalized fractional derivative of Caputo for $0<\alpha <1$ is given by the following expressions [6]:

$${\partial }_t^{\alpha ,\eta }u\left(t\right)={\displaystyle \frac{1}{\Gamma (1-\alpha )}}{\int }_0^t{(t-s)}^{-\alpha }{e}^{-\eta (t-s)}{u}_s\left(s\right)ds,0\le \eta ,$$

in which

$${\partial }_t^{\alpha ,\eta }u\left(t\right)=I^{1-\alpha ,\eta }{u}_t\left(t\right),$$

(2)

and

$$I^{\alpha ,\eta }u\left(t\right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_0^t{(t-s)}^{\alpha -1}{e}^{-\eta (t-s)}u\left(s\right)ds,0\le \eta ,$$

in which $\Gamma$ and $I^{\alpha ,\eta }$ are the Euler gamma function and the operator of the exponential fractional integro-differential, respectively.

In this paper, we are interested in the global existence and general decay phenomenon and we investigate the relationship between the logarithmic source interaction and the behavior of solutions with the presence of the delay at the same time coupling by the acoustic and fractional boundary conditions. This general topic has attracted a lot of attention in recent years. Regarding the source logarithm, we find it in many different works, including fields and phenomena in applied sciences. For greater depth, we refer the reader to the following works [7,8,9,10,11].

The same applies to the delay that we find in most natural phenomena and applied sciences as well. This has also received the attention of many researchers, as some results have been published regarding stability, the presence of a solution, detonation, etc. What follows can be found in this context [12,13,14]. Caputo’s fractional derivatives have undergone significant development over the past few decades, supported by a multitude of definitions and results in the literature. Notable contributions to this field can be found in works such as [15,16,17].

Conversely, the acoustic boundary conditions introduced by Morse and Ingard in their work [18] have given rise to numerous challenges. Subsequent researchers have expanded upon this foundation, yielding significant results in the field. We encourage readers to consult the following studies for a more in-depth understanding: [18,19]. Additionally, for issues related to boundary dissipation, see [20,21,22,23]. Based on these contributions, the combination of various damping and source terms presents a novel problem distinct from those previously studied. The objective of this study is to establish the global existence and general decay of solutions for a wave equation that incorporates acoustic and fractional boundary conditions, along with logarithmic source and delay terms, under suitable assumptions derived from our findings.

This concept is divided into several sections. We provide some basic definitions and notions that are needed for the analysis of the problem in Section 2. The third section is concerned with studying the global existence of the solution. After that, the general decay is described in the next section. Finally, we provide a general summary with future goals for this work.

2. Theory and Concepts

In this section, we introduce the notations, lemmas, and assumptions that are fundamental to the proof of our results.

First, consider

$$H_{{\Sigma }_1}^1(\Omega )=\left\{u\in H^1(\Omega ),u=0\mathrm{on}{\Sigma }_1=0\right\}.$$

Lemma 1

(The Sobolev–Poincaré inequality. See [24]). Take a number q in a way such that $1\le q\le +\infty$ $\left(N=1,2\right)$ or $1\le q\le {\displaystyle \frac{N+2}{N-2}}$, $\left(N\ge 3\right)$. Then, $\exists C_{\ast }>0$ satisfying

$${\parallel u\parallel }_{q+1}\le C_{\ast }{\parallel \nabla u\parallel }_2,\forall u\in H_0^1(\Omega ).$$

Lemma 2

(Trace Sobolev embedding. See [25,26]). C1-In case

$$1\le p\le {\displaystyle \frac{N+2}{N-2}},(N\ge 3)\mathit{or}1\le p\le \infty ,(N=1,2),$$

(3)

then

$$H_{{\Sigma }_1}^1(\Omega )\hookrightarrow L^{p+1}(\Omega ),$$

i.e.,

$${\parallel u\parallel }_{p+1}\le B_{p,\Omega }{\parallel \nabla u\parallel }_2,\forall u\in H_{{\Sigma }_1}^1(\Omega ).$$

C2-In case

$$1\le q\le {\displaystyle \frac{N}{N-2}},(N\ge 3)\mathit{or}1\le q\le \infty ,(N=1,2),$$

(4)

then

$$H_{{\Sigma }_1}^1(\Omega )\hookrightarrow L^{q+1}\left({\Sigma }_0\right),$$

i.e.,

$${\parallel u\parallel }_{q+1,{\Sigma }_0}\le B_{q,{\Sigma }_0}{\parallel \nabla u\parallel }_2,\forall u\in H_{{\Sigma }_1}^1(\Omega ).$$

In which the best constants that satisfy the trace Sobolev embedding are $B_{p,\Omega }$ and $B_{q,{\Sigma }_0}$.

Theorem 1

([23], Theorem 1). Introduce μ as follows:

$$\mu \left(\xi \right)={\left|\xi \right|}^{{\displaystyle \frac{(2\alpha -1)}{2}}},0<\alpha <1,\xi \in \mathbb{R}.$$

(5)

Hence, we find

$$O=I^{1-\alpha ,\eta }U.$$

(6)

We conclude that

$${\partial }_t\varphi (\xi ,t)+({\xi }^2+\eta )\varphi (\xi ,t)-U(L,t)\mu \left(\xi \right)=0,0<t,0\le \eta ,$$

(7)

$$\varphi (\xi ,0)=0,\xi \in \mathbb{R},$$

(8)

$$O\left(t\right)={\displaystyle \frac{sin\left(\alpha \pi \right)}{\pi }}{\int }_{-\infty }^{+\infty }\varphi (\xi ,t)\mu \left(\xi \right)d\xi ,0<t,\xi \in \mathbb{R},.$$

(9)

To arrive at our objective, we have these necessary suppositions for $\mathfrak{P},\mathfrak{Q}$, and ${\mathfrak{A}}_2$:

(H1)

$$\exists p_i,q_i>0(i=1,2),\mathrm{such}\mathrm{that}{p}_0\le \mathfrak{P}\left(x\right)\le p_1,q_0\le \mathfrak{Q}\left(x\right)\le q_1,\forall x\in {\Gamma }_0.$$

(10)

(H2)

$$|{\mathfrak{A}}_2|<{\mathfrak{A}}_1.$$

(11)

At this moment, as in [14], we pick the following novel variables:

$$z(x,\varpi ,t)=u_t(x,t-\tau \varpi ),(x,\varpi ,t)\in \Omega \times (0,1)\times {\mathbb{R}}_{+},$$

satisfying

$$\left\{\begin{array}{c}\tau z_t(x,\varpi ,t)+z_{\rho }(x,\varpi ,t)=0\hfill \\ z(x,0,t)=u_t(x,t).\hfill \end{array}\right.$$

(12)

Hence, the Theorem 1 and (2) gives

$$\left\{\begin{array}{cc}{u}_{tt}-\Delta u+{\mathfrak{A}}_1{u}_t+{\mathfrak{A}}_{2}z(x,1,t)={\left|u\right|}^{p-2}uln\left|u\right|,\hfill & x\in \Omega ,0<t,\hfill \\ {\partial }_t\varphi (\xi ,t)+({\xi }^2+\eta )\varphi (\xi ,t)-u_t(x,t)\mu \left(\xi \right)=0,\hfill & x\in {\Sigma }_0,\xi \in \mathbb{R},0<t,\hfill \\ \tau z_t(x,\varpi ,t)+z_{\varpi }(x,\varpi ,t)=0\hfill & (x,\varpi ,t)\in \Omega \times (0,1)\times {\mathbb{R}}_{+}\hfill \\ {\displaystyle \frac{\partial u}{\partial \nu }}=-{\mathfrak{B}}_1{\int }_{-\infty }^{+\infty }\varphi (\xi ,t)\mu \left(\xi \right)d\xi +{\chi }_t,\hfill & x\in {\Sigma }_0,\xi \in \mathbb{R},0<t,\hfill \\ u_t+\mathfrak{P}\left(x\right){\chi }_t+\mathfrak{A}Q\left(x\right)\chi =0,\hfill & x\in {\Sigma }_0,0<t,\hfill \\ u=0,\hfill & x\in {\Sigma }_1,t>0,\hfill \\ u(x,0)=u_0\left(x\right),u_t(x,0)=u_1\left(x\right),\hfill & x\in \Omega ,\hfill \\ \varphi (\xi ,0)=0,\hfill & \xi \in \mathbb{R},\hfill \\ \chi (x,0)={\chi }_0,\hfill & x\in {\Sigma }_0,\hfill \\ z(x,\varpi ,0)=f_0(x,-\tau \varpi ),\hfill & (x,\varpi )\in \Omega \times (0,1).\hfill \end{array}\right.$$

(13)

By arguments [9,10,27], we give the following well-posed result of the problem (13) without proof.

Proposition 1.

Suppose (10) and (11) holds. Then, for all  $(u_0,u_1,{\varphi }_0,f_0,{\chi }_0)\in \mathcal{H}$ , then (13) has a unique solution, for some $T>0$:

$$\begin{array}{ccc}& & u\in C([0,T];H^2(\Omega )\cap H_{{\Sigma }_1}^1(\Omega )),\hfill \\ & & u_t\in C([0,T];H_{{\Sigma }_1}^1(\Omega ))\cap L^2(\Omega \times (0,1)),\hfill \\ & & \varphi \in C([0,T];L^2\left(\mathbb{R}\right)),\hfill \\ & & \chi ,{\chi }_t\in L^2({\mathbb{R}}_{+},L^2\left({\Sigma }_0\right)).\hfill \end{array}$$

where

$$\mathcal{H}=H_{{\Sigma }_1}^1(\Omega )\times L^2(\Omega )\times L^2\left(\mathbb{R}\right)\times L^2(\Omega \times (0,1))\times L^2\left({\Sigma }_0\right).$$

Next, we give the functional E of the energy for the system (13).

Lemma 3.

Consider $(u,\varphi ,z,\chi )$ as the regular solution of (13). Then, the functional of energy is defined by

$$\begin{array}{ccc}\hfill E\left(t\right)& =& {\displaystyle \frac{1}{2}}\parallel u_t{\parallel }_2^2+{\displaystyle \frac{1}{2}}{\parallel \nabla u\parallel }_2^2+{\displaystyle \frac{1}{p^2}}{\parallel u\parallel }_p^p-{\displaystyle \frac{1}{p}}{\int }_{\Omega }ln\left|u\right|u^{p}dx+{\displaystyle \frac{1}{2}}{\int }_{{\Sigma }_0}\mathfrak{Q}\left(x\right){\chi }^{2}d\Sigma \hfill \\ & & +{\displaystyle \frac{{\mathfrak{B}}_1}{2}}{\int }_{{\Sigma }_0}{\int }_{-\infty }^{+\infty }{\left|\varphi (\xi ,t)\right|}^{2}d\xi d\Sigma +{\displaystyle \frac{\Theta }{2}}{\int }_0^1{\parallel z\left(x,\varpi ,t\right)\parallel }_2^{2}d\varpi ,\hfill \end{array}$$

(14)

satisfying

$$\begin{array}{ccc}\hfill E^{\prime }\left(t\right)& \le & -C_0\left(\parallel u_t{\parallel }_2^2+{\parallel z\left(x,1,t\right)\parallel }_2^2\right)-{\int }_{{\Sigma }_0}\mathfrak{P}\left(x\right){\chi }_t^{2}d\Sigma \hfill \\ & & -{\mathfrak{B}}_1{\int }_{{\Sigma }_0}{\int }_{-\infty }^{+\infty }({\xi }^2+\eta ){\left|\varphi (\xi ,t)\right|}^{2}d\xi d\Sigma \le 0,\hfill \end{array}$$

(15)

for $\Theta >0$, verifying

$$\tau |{\mathfrak{A}}_2|<\Theta <\tau (2{\mathfrak{A}}_1-|{\mathfrak{A}}_2\left|\right).$$

(16)

Proof. 

First of all, multiplying the Equation (13) by $u_t$, we find

$$\begin{array}{c}\hfill {\int }_{\Omega }{u}_{tt}{u}_t-{\int }_{\Omega }\Delta u{u}_{t}dx+{\mathfrak{A}}_2{\int }_{\Omega }{u}_{t}z(x,1,t)dx+{\mathfrak{A}}_1\parallel u_t{\parallel }_2^2={\int }_{\Omega }{\left|u\right|}^{p-2}uln\left|u\right|u_{t}dx.\end{array}$$

Then, the integration by parts over $\Omega$ gives

$$\begin{array}{ccc}& & {\displaystyle \frac{d}{dt}}\left[{\displaystyle \frac{1}{2}}{\parallel u_t\parallel }_2^2\right.\left.+{\displaystyle \frac{1}{2}}{\parallel \nabla u\parallel }_2^2-{\displaystyle \frac{1}{p}}{\int }_{\Omega }ln\left|u\right|u^{p}dx+{\displaystyle \frac{1}{p^2}}{\parallel u\parallel }_p^p\right]\hfill \\ & & +{\mathfrak{A}}_1{\parallel u_t\parallel }_2^2+{\mathfrak{A}}_2{\int }_{\Omega }{u}_{t}z(x,1,t)dx-{\int }_{{\Sigma }_0}{u}_t{\chi }_{t}d\Sigma \hfill \\ & & +{\mathfrak{B}}_1{\int }_{{\Sigma }_0}{u}_t(x,t){\int }_{-\infty }^{+\infty }\mu \left(\xi \right)\varphi (\xi ,t)d\xi d\Sigma =0.\hfill \end{array}$$

(17)

Next, multiply ${\chi }_t$ by (13). When we combine the outcome with the prior equality (17) after integrating over ${\Sigma }_0$, we obtain

$$\begin{array}{ccc}& & {\displaystyle \frac{d}{dt}}\left[{\displaystyle \frac{1}{2}}{\parallel u_t\parallel }_2^2\right.\left.+{\displaystyle \frac{1}{2}}{\parallel \nabla u\parallel }_2^2-{\displaystyle \frac{1}{p}}{\int }_{\Omega }ln\left|u\right|u^{p}dx+{\displaystyle \frac{1}{p^2}}{\parallel u\parallel }_p^p+{\displaystyle \frac{1}{2}}{\int }_{{\Sigma }_0}\mathfrak{Q}\left(x\right){\chi }^{2}d\Sigma \right]\hfill \\ & & +{\mathfrak{A}}_1{\parallel u_t\parallel }_2^2+{\mathfrak{A}}_2{\int }_{\Omega }{u}_{t}z(x,1,t)dx+{\int }_{{\Sigma }_0}\mathfrak{P}\left(x\right){\chi }_t^{2}d\Sigma \hfill \\ & & +{\mathfrak{B}}_1{\int }_{{\Sigma }_0}{u}_t(x,t){\int }_{-\infty }^{+\infty }\mu \left(\xi \right)\varphi (\xi ,t)d\xi d\Sigma =0.\hfill \end{array}$$

(18)

Secondly, multiplying the Equation (13) by ${\mathfrak{B}}_1\varphi$ and integrating over ${\Sigma }_0\times (-\infty ,+\infty )$, one finds

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\mathfrak{B}}_1}{2}}{\displaystyle \frac{d}{dt}}{\int }_{{\Sigma }_0}{\int }_{-\infty }^{+\infty }& {\left|\varphi (\xi ,t)\right|}^{2}d\xi d\Sigma +{\mathfrak{B}}_1{\int }_{{\Sigma }_0}{\int }_{-\infty }^{+\infty }({\xi }^2+\eta ){\left|\varphi (\xi ,t)\right|}^{2}d\xi d\Sigma \hfill \\ \hfill & -{\mathfrak{B}}_1{\int }_{{\Sigma }_0}{u}_t(x,t){\int }_{-\infty }^{+\infty }\mu \left(\xi \right)\varphi (\xi ,t)d\xi d\Sigma =0.\hfill \end{array}$$

(19)

In the third stage, multiplying (13) by ${\displaystyle \frac{\Theta }{\tau }}z$, integrating over $\Omega \times (0,1)$ and applying (12), one finds

$$\begin{array}{ccc}\hfill {\displaystyle \frac{d}{dt}}{\displaystyle \frac{\Theta }{2}}{\int }_{\Omega }{\int }_0^1{z}^2(x,\varpi ,t)d\varpi dx& =& -{\displaystyle \frac{\Theta }{\tau }}{\int }_{\Omega }{\int }_0^{1}z{z}_{\varpi }\left(x,\varpi ,t\right)d\varpi dx\hfill \\ & =& -{\displaystyle \frac{\Theta }{2\tau }}{\int }_{\Omega }{\int }_0^1{\displaystyle \frac{d}{d\varpi }}{z}^2\left(x,\varpi ,t\right)d\varpi dx\hfill \\ & =& {\displaystyle \frac{\Theta }{2\tau }}{\int }_{\Omega }(z^2\left(x,0,t\right)-z^2(x,1,t))dx\hfill \\ & =& {\displaystyle \frac{\Theta }{2\tau }}\parallel u_t{\parallel }_2^2-{\displaystyle \frac{\Theta }{2\tau }}{\parallel z(x,1,t)\parallel }_2^2,\hfill \end{array}$$

(20)

and by Young’s inequality, we have

$$\begin{array}{c}\hfill {\mathfrak{A}}_2{\int }_{\Omega }{u}_{t}z(x,1,t)dx\le {\displaystyle \frac{|{\mathfrak{A}}_2|}{2}}\parallel u_t{\parallel }_2^2+{\displaystyle \frac{|{\mathfrak{A}}_2|}{2}}{\parallel z(x,1,t)\parallel }_2^2.\end{array}$$

(21)

From (18), (19), (20) and (21), we obtain (14) and

$$\begin{array}{ccc}\hfill {\displaystyle \frac{d}{dt}}E\left(t\right)& =& -\left({\mathfrak{A}}_1-{\displaystyle \frac{\Theta }{2\tau }}-{\displaystyle \frac{|{\mathfrak{A}}_2|}{2}}\right)\parallel u_t{\parallel }_2^2-\left({\displaystyle \frac{\Theta }{2\tau }}-{\displaystyle \frac{|{\mathfrak{A}}_2|}{2}}\right){\parallel z(x,1,t)\parallel }_2^2\hfill \\ & & -{\int }_{{\Sigma }_0}\mathfrak{P}\left(x\right){\chi }_t^{2}d\Sigma -{\mathfrak{B}}_1{\int }_{{\Sigma }_0}{\int }_{-\infty }^{+\infty }({\xi }^2+\eta ){\left|\varphi (\xi ,t)\right|}^{2}d\xi d\Sigma \le 0.\hfill \end{array}$$

Keeping in mind condition (16), we observe that

$${\mathfrak{A}}_3=\left({\mathfrak{A}}_1-{\displaystyle \frac{\Theta }{2\tau }}-{\displaystyle \frac{|{\mathfrak{A}}_2|}{2}}\right)>0,\mathrm{and}{\mathfrak{A}}_4=\left({\displaystyle \frac{\Theta }{2\tau }}-{\displaystyle \frac{|{\mathfrak{A}}_2|}{2}}\right)>0.$$

(22)

Then, by (16), (22), (11), we obtain (15), where $C_0=min\{{\mathfrak{A}}_3,{\mathfrak{A}}_4\}>0$.

• Consequently,

  $$E\left(t\right)\le E\left(0\right).$$

  (23)

□

3. Global Existence

This unit of work is devoted to proving the global existence of weak solutions for our problem (13). Our plan for this purpose is to prove that the solution of (13) is uniformly bounded and global in time. For this objective, we propose the following functionals:

$$J\left(t\right)={\displaystyle \frac{1}{2}}{\parallel \nabla u\parallel }_2^2-{\displaystyle \frac{1}{p}}{\int }_{\Omega }ln\left|u\right|u^{p}dx+{\displaystyle \frac{1}{p^2}}{\parallel u\parallel }_p^p,$$

(24)

and

$$I\left(t\right)={\parallel \nabla u\parallel }_2^2-{\int }_{\Omega }ln\left|u\right|u^{p}dx.$$

(25)

Based on the previous definitions of functions $I\left(t\right),J\left(t\right)$ and $E\left(t\right)$, we find

$$J\left(t\right)={\displaystyle \frac{1}{p}}I\left(t\right)+\left({\displaystyle \frac{p-2}{2p}}\right){\parallel \nabla u\parallel }_2^2+{\displaystyle \frac{1}{p^2}}{\parallel u\parallel }_p^p,$$

(26)

and

$$\begin{array}{ccc}\hfill E\left(t\right)& =& {\displaystyle \frac{1}{2}}\parallel u_t{\parallel }_2^2+{\displaystyle \frac{{\mathfrak{B}}_1}{2}}{\int }_{{\Sigma }_0}{\int }_{-\infty }^{+\infty }{\left|\varphi (\xi ,t)\right|}^{2}d\xi d\Sigma +{\displaystyle \frac{\Theta }{2}}{\int }_0^1{\parallel z\left(x,\varpi ,t\right)\parallel }_2^{2}d\varpi \hfill \\ & & +{\displaystyle \frac{1}{2}}{\int }_{{\Sigma }_0}\mathfrak{Q}\left(x\right){\chi }^{2}d\Sigma +J\left(t\right)\hfill \\ & =& {\displaystyle \frac{1}{2}}\parallel u_t{\parallel }_2^2+{\displaystyle \frac{{\mathfrak{B}}_1}{2}}{\int }_{{\Sigma }_0}{\int }_{-\infty }^{+\infty }{\left|\varphi (\xi ,t)\right|}^{2}d\xi d\Sigma +{\displaystyle \frac{\Theta }{2}}{\int }_0^1{\parallel z\left(x,\varpi ,t\right)\parallel }_2^{2}d\varpi \hfill \\ & & +{\displaystyle \frac{1}{2}}{\int }_{{\Sigma }_0}\mathfrak{Q}\left(x\right){\chi }^{2}d\Sigma +{\displaystyle \frac{1}{p}}I\left(t\right)+\left({\displaystyle \frac{p-2}{2p}}\right){\parallel \nabla u\parallel }_2^2+{\displaystyle \frac{1}{p^2}}{\parallel u\parallel }_p^p.\hfill \end{array}$$

(27)

Lemma 4.

Suppose that the initial data $(u_0,u_1,{\varphi }_0,f_0,{\chi }_0)\in (H_{{\Sigma }_1}^1(\Omega )\times L^2(\Omega )\times L^2\left(\mathbb{R}\right)\times L^2(\Omega \times (0,1))\times L^2\left({\Sigma }_0\right))$, satisfying $I\left(0\right)>0$ and

$$\zeta :=B_{p,\Omega }^{p+1}{\left({\displaystyle \frac{2p}{p-1}}E\left(0\right)\right)}^{{\displaystyle \frac{p-1}{2}}}<1.$$

(28)

This implies that for all $t\in [0,T]$, $I\left(t\right)>0$.

Proof. 

Based on the inequality $I\left(0\right)>0$, by continuity, we conclude that $\exists T^{\ast },0<T^{\ast }\le T$ in a manner that for all $t\in [0,T^{\ast }]$, $I\left(t\right)\ge 0$. Thus, we have

$$\begin{array}{ccc}\hfill J\left(t\right)& \ge & {\displaystyle \frac{1}{p}}I\left(t\right)+\left({\displaystyle \frac{p-2}{2p}}\right){\parallel \nabla u\parallel }_2^2+{\displaystyle \frac{1}{p^2}}{\parallel u\parallel }_p^p\hfill \\ & \ge & \left({\displaystyle \frac{p-2}{2p}}\right){\parallel \nabla u\parallel }_2^2\forall t\in [0,T^{\ast }].\hfill \end{array}$$

(29)

Then, we use (23) and (27) and find

$$\begin{array}{ccc}\hfill {\parallel \nabla u\parallel }_2^2& \le & {\displaystyle \frac{2p}{p-2}}J\left(t\right)\le {\displaystyle \frac{2p}{p-2}}E\left(t\right)\hfill \\ & \le & {\displaystyle \frac{2p}{p-2}}E\left(0\right),\forall t\in [0,T^{\ast }].\hfill \end{array}$$

(30)

Furthermore, the embedding $H_0^1(\Omega ))\hookrightarrow L^{p+1}(\Omega )$ gives

$$\begin{array}{ccc}\hfill {\int }_{\Omega }ln\left|u\right|u^{p}dx& \le & {\parallel u\parallel }_{p+1}^{p+1}\hfill \\ & \le & B_{p,\Omega }^{p+1}{\parallel \nabla u\parallel }_2^{p+1}.\hfill \end{array}$$

(31)

By (30), we find

$$\begin{array}{ccc}\hfill {\int }_{\Omega }ln\left|u\right|u^{p}dx& <& B_{p,\Omega }^{p+1}{\left({\displaystyle \frac{2pE\left(0\right)}{(p-2)}}\right)}^{{\displaystyle \frac{p-1}{2}}}{\parallel \nabla u\left(t\right)\parallel }_2^2\hfill \\ & <& {\zeta \parallel \nabla u\left(t\right)\parallel }_2^2,\hfill \end{array}$$

(32)

where

$$\zeta :=B_{p,\Omega }^{p+1}{\left({\displaystyle \frac{2pE\left(0\right)}{(p-2)}}\right)}^{{\displaystyle \frac{p-1}{2}}}.$$

According to (25), (28) and (32), we obtain

$$\begin{array}{ccc}\hfill I\left(t\right)& >& {(1-\zeta )\parallel \nabla u\left(t\right)\parallel }_2^2>0,\forall t\in [0,T^{\ast }].\hfill \end{array}$$

(33)

This method can be repeated to extend $T^{\ast }$ to T. This brings the proof to an end. □

Remark 1.

The hypotheses of Lemma 4 gives $0\le J\left(t\right)$, which implies that for all $t\in [0,T]$, $E\left(t\right)\ge 0$. Then, the definition (27) and the estimate (29) gives

$$\begin{array}{ccc}\hfill \parallel u_t{\left(t\right)\parallel }_2^2& \le & 2E\left(0\right),\hfill \\ \hfill {\parallel u\left(t\right)\parallel }_p^p& \le & p^{2}E\left(0\right),\hfill \\ \hfill {\int }_{{\Sigma }_0}{\int }_{-\infty }^{+\infty }{\left|\varphi (\xi ,t)\right|}^{2}d\xi d\Sigma & \le & {\displaystyle \frac{2}{{\mathfrak{B}}_1}}E\left(0\right),\hfill \\ \hfill {\int }_0^1{\parallel z\left(x,\varpi ,t\right)\parallel }_2^{2}d\varpi & \le & {\displaystyle \frac{2}{\Theta }}E\left(0\right),\hfill \\ \hfill {\int }_{{\Sigma }_0}\mathfrak{Q}\left(x\right){\chi }^{2}d\Sigma & \le & 2E\left(0\right).\hfill \end{array}$$

(34)

Theorem 2.

The solution to (13) is global and bounded if the conditions of Lemma 4 are satisfied.

Proof. 

It is enough to prove the time-independent boundedness of

$$\begin{array}{ccc}\hfill {\parallel (u,\varphi ,z,\chi )\parallel }_H& :=& \parallel u_t{\parallel }_2^2+{\parallel \nabla u\parallel }_2^2+{\parallel u\parallel }_p^p+{\int }_{{\Sigma }_0}{\int }_{-\infty }^{+\infty }{\left|\varphi (\xi ,t)\right|}^{2}d\xi d\Sigma \hfill \\ & & +{\int }_0^1\parallel z\left(x,\varpi ,t\right){\parallel }_2^{2}d\varpi +{\parallel \chi \parallel }_{{\Sigma }_0}^2.\hfill \end{array}$$

To achieve this, we utilize (34) and obtain the following:

$$\begin{array}{ccc}\hfill E\left(0\right)>E\left(t\right)& =& J\left(t\right)+{\displaystyle \frac{1}{2}}\parallel u_t{\parallel }_2^2+{\displaystyle \frac{{\mathfrak{B}}_1}{2}}{\int }_{{\Sigma }_0}{\int }_{-\infty }^{+\infty }{\left|\varphi (\xi ,t)\right|}^{2}d\xi d\Sigma \hfill \\ & & +{\displaystyle \frac{\Theta }{2}}{\int }_0^1{\parallel z\left(x,\varpi ,t\right)\parallel }_2^{2}d\varpi +{\displaystyle \frac{1}{2}}{\int }_{{\Sigma }_0}\mathfrak{Q}\left(x\right){\chi }^{2}d\Sigma \hfill \\ & \ge & {\displaystyle \frac{1}{2}}\parallel u_t{\parallel }_2^2+{\displaystyle \frac{{\mathfrak{B}}_1}{2}}{\int }_{{\Sigma }_0}{\int }_{-\infty }^{+\infty }{\left|\varphi (\xi ,t)\right|}^{2}d\xi d\Sigma +{\displaystyle \frac{\Theta }{2}}{\int }_0^1{\parallel z\left(x,\varpi ,t\right)\parallel }_2^{2}d\varpi \hfill \\ & & +\left({\displaystyle \frac{p-2}{2p}}\right){\parallel \nabla u\parallel }_2^2+{\displaystyle \frac{1}{p^2}}{\parallel u\parallel }_p^p+{\displaystyle \frac{q_0}{2}}{\parallel \chi \parallel }_{{\Sigma }_0}^2.\hfill \end{array}$$

(35)

Hence,

$${\parallel (v,\varphi ,z,\chi )\parallel }_H\le CE\left(0\right),$$

where $C(p,\Theta ,{\mathfrak{B}}_1,q_0)>0$. □

4. Decay of Solutions

This unit is investigate with the proof of the exponential decay result of the system (13). To attain our objective, we use the definition of the following functionals for $N>0$:

$$\mathcal{L}\left(t\right)=NE\left(t\right)+\left({\Psi }_1\left(t\right)+{\displaystyle \frac{b_1}{2}}{\Psi }_2\left(t\right)\right)+{\Psi }_3\left(t\right),$$

(36)

and

$$\begin{array}{cc}\hfill {\Psi }_1\left(t\right)& ={\int }_{\Omega }{u}_{t}udx+{\int }_{{\Sigma }_0}u\chi d\Sigma +{\displaystyle \frac{1}{2}}{\int }_{{\Sigma }_0}\mathfrak{P}\left(x\right){\chi }^{2}d\Sigma +{\displaystyle \frac{{\mathfrak{A}}_1}{2}}{\int }_{\Omega }{u}^{2}dx,\hfill \\ \hfill {\Psi }_2\left(t\right)& ={\int }_{{\Sigma }_0}{\int }_{-\infty }^{+\infty }{\displaystyle \frac{1}{{\xi }^2+\eta }}{{\rm Y}}^2(x,\xi ,t)d\xi d\Sigma ,\hfill \\ \hfill {\Psi }_3\left(t\right)& ={\displaystyle \frac{1}{2}}{\int }_{\Omega }{\int }_0^1{e}^{-2\tau \varpi }{z}^2\left(x,\varpi ,t\right)d\varpi dx,\hfill \end{array}$$

(37)

where

$${\rm Y}(x,\xi ,t):=({\xi }^2+\eta ){\int }_0^t\varphi (\xi ,s)ds+u_0\left(x\right)\mu \left(\xi \right).$$

(38)

Lemma 5.

Consider $(u,\varphi ,z,\chi )$ as the solution of (13). Hence, we obtain

$$\begin{array}{ccc}\hfill {\int }_{{\Sigma }_0}{\int }_{-\infty }^{+\infty }\varphi (\xi ,t){\rm Y}(x,\xi ,t)d\xi d\Sigma & =& {\int }_{{\Sigma }_0}u(x,t){\int }_{-\infty }^{+\infty }\varphi (\xi ,t)\mu \left(\xi \right)d\xi d\Sigma \hfill \\ & & -{\int }_{{\Sigma }_0}{\int }_{-\infty }^{+\infty }{\left|\varphi (\xi ,t)\right|}^{2}d\xi d\Sigma .\hfill \end{array}$$

Proof. 

From (13), we have

$$({\xi }^2+\eta )\varphi (\xi ,t)=u_t(x,t)\mu \left(\xi \right)-{\partial }_t\varphi (\xi ,t),\forall x\in {\Sigma }_0.$$

(39)

Based on the integration of (39) over $(0,t)$, and by (13), we obtain

$${\int }_0^t({\xi }^2+\eta )\varphi (\xi ,s)ds=u(x,t)\mu \left(\xi \right)-u_0\left(x\right)\mu \left(\xi \right)-\varphi (\xi ,t),\forall x\in {\Sigma }_0,$$

(40)

then,

$${\rm Y}(x,\xi ,t)=u(x,t)\mu \left(\xi \right)-\varphi (\xi ,t),\forall x\in {\Sigma }_0.$$

(41)

After simplification, we obtain

$$\begin{array}{ccc}\hfill {\int }_{{\Sigma }_0}{\int }_{-\infty }^{+\infty }\varphi (\xi ,t){\rm Y}(x,\xi ,t)d\xi d\Sigma & =& {\int }_{{\Sigma }_0}u(x,t){\int }_{-\infty }^{+\infty }\varphi (\xi ,t)\mu \left(\xi \right)d\xi d\Sigma \hfill \\ & & -{\int }_{{\Sigma }_0}{\int }_{-\infty }^{+\infty }{\left|\varphi (\xi ,t)\right|}^{2}d\xi d\Sigma .\hfill \end{array}$$

□

Next, by definition of the functionals ${\Psi }_1,{\Psi }_2$ and ${\Psi }_3$ in (37), the following Lemmas are obtained.

Lemma 6.

The functional ${\Psi }_1^{\prime }\left(t\right)+{\displaystyle \frac{{\mathfrak{B}}_1}{2}}{\Psi }_2^{\prime }\left(t\right)$ verifies

$$\begin{array}{ccc}\hfill {\Psi }_1^{\prime }\left(t\right)+{\displaystyle \frac{{\mathfrak{B}}_1}{2}}{\Psi }_2^{\prime }\left(t\right)& \le & -(1-2c_{\ast }{\delta }_1){\parallel \nabla u\parallel }_2^2-{\int }_{{\Sigma }_0}\mathfrak{Q}\left(x\right){\chi }^{2}d\Sigma +{\int }_{\Omega }ln\left|u\right|{\left|u\right|}^{p}dx\hfill \\ & & +\parallel u_t{\parallel }_2^2+c\left({\delta }_1\right)\parallel z(x,1,t){\parallel }_2^2+c\left({\delta }_1\right){\parallel {\chi }_t\parallel }_{{\Sigma }_0}^2\hfill \\ & & -{\mathfrak{B}}_1{\int }_{{\Sigma }_0}{\int }_{-\infty }^{+\infty }{\left|\varphi (\xi ,t)\right|}^{2}d\xi d\Sigma .\hfill \end{array}$$

(42)

Proof. 

By direct calculation applying the integration by parts and using Young’s inequality, we find

$$\begin{array}{ccc}\hfill {\Psi }_1^{\prime }\left(t\right)+{\displaystyle \frac{{\mathfrak{B}}_1}{2}}{\Psi }_2^{\prime }\left(t\right)& =& \parallel u_t{\parallel }_2^2+{\int }_{\Omega }{u}_{tt}udx+{\int }_{{\Sigma }_0}{u}_t\chi d\Sigma +{\int }_{{\Sigma }_0}u{\chi }_{t}d\Sigma +{\int }_{{\Sigma }_0}\mathfrak{P}\left(x\right){\chi }_t\chi d\Sigma \hfill \\ & & +{\mathfrak{A}}_1{\int }_{\Omega }u{u}_{t}dx+{\mathfrak{B}}_1{\int }_{{\Sigma }_0}{\int }_{-\infty }^{+\infty }\varphi (\xi ,t){\rm Y}(x,\xi ,t)d\xi d\Sigma \hfill \\ & =& \parallel u_t{\parallel }_2^2-{\parallel \nabla u\parallel }_2^2-{\int }_{{\Sigma }_0}\mathfrak{Q}\left(x\right){\chi }^{2}d\Sigma +{\int }_{\Omega }ln\left|u\right|{\left|u\right|}^{p}dx\hfill \\ & & -{\mathfrak{A}}_2{\int }_{\Omega }uz(x,1,t)dx+2{\int }_{{\Sigma }_0}u{\chi }_{t}d\Sigma \hfill \\ & & -{\mathfrak{B}}_1{\int }_{{\Sigma }_0}u(x,t){\int }_{-\infty }^{+\infty }\mu \left(\xi \right)\varphi (\xi ,t)d\xi d\Sigma \hfill \\ & & +{\mathfrak{B}}_1{\int }_{{\Sigma }_0}{\int }_{-\infty }^{+\infty }\varphi (\xi ,t){\rm Y}(x,\xi ,t)d\xi d\Sigma .\hfill \end{array}$$

(43)

After that, Lemma 5 gives

$$\begin{array}{cc}\hfill {\Psi }_1^{\prime }\left(t\right)+{\displaystyle \frac{{\mathfrak{B}}_1}{2}}{\Psi }_2^{\prime }\left(t\right)& =\parallel u_t{\parallel }_2^2-{\parallel \nabla u\parallel }_2^2-{\int }_{{\Sigma }_0}\mathfrak{Q}\left(x\right){\chi }^{2}d\Sigma +{\int }_{\Omega }ln\left|u\right|{\left|u\right|}^{p}dx\hfill \\ \hfill & -{\mathfrak{B}}_1{\int }_{{\Sigma }_0}{\int }_{-\infty }^{+\infty }{\left|\varphi (\xi ,t)\right|}^{2}d\xi d\Sigma +2{\int }_{{\Sigma }_0}u{\chi }_{t}d\Sigma \hfill \\ \hfill & -{\mathfrak{A}}_2{\int }_{\Omega }uz(x,1,t)dx.\hfill \end{array}$$

(44)

By Young’s and Poincaré’s inequalities, we find for any ${\delta }_1>0$,

$$\begin{array}{ccc}\hfill {\mathfrak{A}}_2{\int }_{\Omega }uz(x,1,t)dx& \le & c\left({\delta }_1\right)\parallel z(x,1,t){\parallel }_2^2+c_{\ast }{\delta }_1{\parallel \nabla u\parallel }_2^2,\hfill \\ \hfill 2{\int }_{{\Sigma }_0}u{\chi }_{t}d\Sigma & \le & c\left({\delta }_1\right)\parallel {\chi }_t{\parallel }_{{\Sigma }_0}^2+c_{\ast }{\delta }_1{\parallel \nabla u\parallel }_2^2.\hfill \end{array}$$

(45)

By substitution of (45) into (44), we obtain (42). □

Lemma 7.

The functional ${\Psi }_3\left(t\right)$ satisfies

$$\begin{array}{ccc}\hfill {\Psi }_3^{\prime }\left(t\right)& \le & -\kappa {\int }_0^1\parallel z\left(x,\varpi ,t\right){\parallel }_2^{2}d\varpi -{\displaystyle \frac{\kappa }{2\tau }}\parallel z\left(x,1,t\right){\parallel }_2^2+{\displaystyle \frac{1}{2\tau }}{\parallel u_t\parallel }_2^2.\hfill \end{array}$$

(46)

Proof. 

After the direct caculation of the derivative of the function ${\Psi }_3$, and using the third equation in (13), we obtain

$$\begin{array}{ccc}\hfill {\Psi }_3^{\prime }\left(t\right)& =& -{\displaystyle \frac{1}{\tau }}{\int }_{\Omega }{\int }_0^1{e}^{-2\tau \varpi }z{z}_{\varpi }\left(x,\varpi ,t\right)d\varpi dx\hfill \\ & =& -{\int }_{\Omega }{\int }_0^1{e}^{-2\tau \varpi }{z}^2\left(x,\varpi ,t\right)d\varpi dx-{\displaystyle \frac{1}{2\tau }}{\int }_{\Omega }{\int }_0^1{\displaystyle \frac{d}{d\varpi }}\left[e^{-2\tau \varpi }{z}^2\left(x,\varpi ,t\right)\right]d\varpi dx.\hfill \end{array}$$

From the equality $z(x,0,t)=u_t(x,t)$, we find

$$\begin{array}{ccc}\hfill {\Psi }_3^{\prime }\left(t\right)& =& -{\Psi }_3\left(t\right)-{\displaystyle \frac{e^{-2\tau }}{2\tau }}\parallel z\left(x,1,t\right){\parallel }_2^2+{\displaystyle \frac{1}{2\tau }}{\parallel u_t\parallel }_2^2.\hfill \end{array}$$

Hence, by selecting $\kappa =e^{-2\tau }>0$, we obtain (46). □

This unit is investigated with the result of the exponential decay of the global solution.

Theorem 3.

Suppose that the conditions of Lemma 4 $\exists k_i>0,i=1,2$ in a way that the global solution of (13) verifies

$$E\left(t\right)\le k_2{e}^{-k_{1}t}.$$

(47)

Proof. 

After the direct calculation of the derivative of the function (36), we obtain

$$\begin{array}{cc}\hfill {\mathcal{L}}^{\prime }\left(t\right)& =N{E}^{\prime }\left(t\right)+{\Psi }_1^{\prime }\left(t\right)+{\displaystyle \frac{{\mathfrak{B}}_1}{2}}{\Psi }_2^{\prime }\left(t\right)+{\Psi }_3^{\prime }\left(t\right).\hfill \end{array}$$

(48)

From (15), (42) and (46), we have for $\mathfrak{M}>0$

$$\begin{array}{cc}\hfill {\mathcal{L}}^{\prime }\left(t\right)\le & -\left[C_{0}N-c-{\displaystyle \frac{1}{2\tau }}\right]\parallel u_t{\parallel }_2^2-(1-2c_{\ast }{\delta }_1){\parallel \nabla u\parallel }_2^2-(N{p}_0-c\left({\delta }_1\right)){\parallel {\chi }_t\parallel }_{{\Sigma }_0}^2\hfill \\ \hfill & -N{\mathfrak{B}}_1{\int }_{{\Sigma }_0}{\int }_{-\infty }^{+\infty }({\xi }^2+\eta ){\left|\varphi (\xi ,t)\right|}^{2}d\xi d\Sigma +\mathfrak{M}E\left(t\right)-\mathfrak{M}E\left(t\right)\hfill \\ \hfill & -{\int }_{{\Sigma }_0}\mathfrak{Q}\left(x\right){\chi }^{2}d\Sigma -{\mathfrak{B}}_1{\int }_{{\Sigma }_0}{\int }_{-\infty }^{+\infty }{\left|\varphi (\xi ,t)\right|}^{2}d\xi d\Sigma +{\int }_{\Omega }ln\left|u\right|{\left|u\right|}^{p}dx\hfill \\ \hfill & -\kappa {\int }_0^1{\parallel z(x,\varpi ,t)\parallel }_2^{2}d\varpi -(C_{0}N-c\left({\delta }_1\right)-{\displaystyle \frac{\kappa }{2\tau }}){\parallel z(x,1,t)\parallel }_2^2.\hfill \end{array}$$

(49)

Next, by adding and subtracting the energy functional (14), we obtain

$$\begin{array}{cc}\hfill {\mathcal{L}}^{\prime }\left(t\right)\le & -\left[C_{0}N-c-{\displaystyle \frac{1}{2\tau }}-{\displaystyle \frac{\mathfrak{M}}{2}}\right]\parallel u_t{\parallel }_2^2-\left[(1-2c_{\ast }{\delta }_1)-{\displaystyle \frac{\mathfrak{M}}{2}}\right]{\parallel \nabla u\parallel }_2^2\hfill \\ \hfill & -N{\mathfrak{B}}_1{\int }_{{\Sigma }_0}{\int }_{-\infty }^{+\infty }({\xi }^2+\eta )|\varphi (\xi ,t){|}^{2}d\xi d\Sigma +{\displaystyle \frac{\mathfrak{M}}{p^2}}{\parallel u\parallel }_p^p-\mathfrak{M}E\left(t\right)\hfill \\ \hfill & -\left[{\mathfrak{B}}_1-{\displaystyle \frac{\mathfrak{M}}{2}}\right]{\int }_{{\Sigma }_0}{\int }_{-\infty }^{+\infty }{\left|\varphi (\xi ,t)\right|}^{2}d\xi d\Sigma +\left[1-{\displaystyle \frac{\mathfrak{M}}{p}}\right]{\int }_{\Omega }ln\left|u\right|{\left|u\right|}^{p}dx\hfill \\ \hfill & -\left[\kappa -{\displaystyle \frac{\Theta \mathfrak{M}}{2}}\right]{\int }_0^1{\parallel z(x,\varpi ,t)\parallel }_2^{2}d\varpi -\left[C_{0}N-c\left({\delta }_1\right)-{\displaystyle \frac{\kappa }{2\tau }}\right]{\parallel z(x,1,t)\parallel }_2^2\hfill \\ \hfill & -\left[1-{\displaystyle \frac{\mathfrak{M}}{2}}\right]{\int }_{{\Sigma }_0}\mathfrak{Q}\left(x\right){\chi }^{2}d\Sigma -(N{p}_0-c\left({\delta }_1\right)){\parallel {\chi }_t\parallel }_{{\Sigma }_0}^2.\hfill \end{array}$$

(50)

Now, we need to estimate ${\parallel u\parallel }_p^p$ as follows:

$$\begin{array}{cc}\hfill {\parallel u\parallel }_p^p\le B_{p-1,\Omega }^p{\parallel \nabla u\parallel }_2^p& \le B_{p-1,\Omega }^p{\parallel \nabla u\parallel }_2^{{\displaystyle \frac{p-2}{2}}}{\parallel \nabla u\parallel }_2^2\hfill \\ \hfill & \le B_{p-1,\Omega }^p{\left({\displaystyle \frac{2pE\left(0\right)}{p-2}}\right)}^{{\displaystyle \frac{p-2}{2}}}{\parallel \nabla u\parallel }_2^2\hfill \\ \hfill & \le C_B{\parallel \nabla u\parallel }_2^2,\hfill \end{array}$$

(51)

and

$$\begin{array}{cc}\hfill {\int }_{\Omega }ln\left|u\right|{\left|u\right|}^{p}dx\le {\parallel u\parallel }_{p+1}^{p+1}& \le B_{p,\Omega }^{p+1}{\parallel \nabla u\parallel }_2^{{\displaystyle \frac{p-1}{2}}}{\parallel \nabla u\parallel }_2^2\hfill \\ \hfill & \le \left(B_{p,\Omega }^{p+1}/B_{p-1,\Omega }^p\right)C_B{\parallel \nabla u\parallel }_2^2\hfill \\ \hfill & \le {\parallel \nabla u\parallel }_2^2.\hfill \end{array}$$

(52)

According to (50), (51) and (52), we have

$$\begin{array}{cc}\hfill {\mathcal{L}}^{\prime }\left(t\right)\le & -\left[C_{0}N-c-{\displaystyle \frac{1}{2\tau }}-{\displaystyle \frac{\mathfrak{M}}{2}}\right]{\parallel u_t\parallel }_2^2-\left[1-{\displaystyle \frac{\mathfrak{M}}{2}}\right]{\int }_{{\Sigma }_0}\mathfrak{Q}\left(x\right){\chi }^{2}d\Sigma \hfill \\ \hfill & -\left[\underset{\widehat{m}}{\underbrace{\left(1-C_B(B_{p,\Omega }^{p+1}/B_{p-1,\Omega }^p)\right)-2c_{\ast }{\delta }_1}}-\mathfrak{M}\left({\displaystyle \frac{1}{2}}+{\displaystyle \frac{C_B}{p^2}}\right)\right]{\parallel \nabla u\parallel }_2^2\hfill \\ \hfill & -N{\mathfrak{B}}_1{\int }_{{\Sigma }_0}{\int }_{-\infty }^{+\infty }({\xi }^2+\eta ){\left|\varphi (\xi ,t)\right|}^{2}d\xi d\Sigma -\mathfrak{M}E\left(t\right)\hfill \\ \hfill & -\left[{\mathfrak{B}}_1-{\displaystyle \frac{\mathfrak{M}}{2}}\right]{\int }_{{\Sigma }_0}{\int }_{-\infty }^{+\infty }{\left|\varphi (\xi ,t)\right|}^{2}d\xi d\Sigma -(N{p}_0-c\left({\delta }_1\right)){\parallel {\chi }_t\parallel }_{{\Sigma }_0}^2\hfill \\ \hfill & -\left[\kappa -{\displaystyle \frac{\Theta \mathfrak{M}}{2}}\right]{\int }_0^1{\parallel z(x,\varpi ,t)\parallel }_2^{2}d\varpi \hfill \\ \hfill & -\left[C_{0}N-c\left({\delta }_1\right)-{\displaystyle \frac{\kappa }{2\tau }}\right]{\parallel z(x,1,t)\parallel }_2^2.\hfill \end{array}$$

(53)

From (28), we have

$$1-C_B(B_{p,\Omega }^{p+1}/B_{p-1,\Omega }^p)=1-B_{p,\Omega }^{p+1}{\left({\displaystyle \frac{2pE\left(0\right)}{p-2}}\right)}^{{\displaystyle \frac{p-2}{2}}}>0.$$

(54)

Here, we choose ${\delta }_1$ to be sufficiently small so that

$$\widehat{m}:=\left(1-C_B(B_{p,\Omega }^{p+1}/B_{p-1,\Omega }^p)\right)-2c_{\ast }{\delta }_1>0,$$

(55)

then, we select $\mathfrak{M}$ small enough such that

$$\begin{array}{c}\hfill {\beta }_1=\widehat{m}-\mathfrak{M}({\displaystyle \frac{1}{2}}+{\displaystyle \frac{C_B}{p^2}})>0,{\beta }_2={\mathfrak{B}}_1-{\displaystyle \frac{\mathfrak{M}}{2}}>0,{\beta }_3=\kappa -{\displaystyle \frac{\mathfrak{M}\Theta }{2}}>0.\end{array}$$

(56)

The estimate (53) gives

$$\begin{array}{cc}\hfill {\mathcal{L}}^{\prime }\left(t\right)\le & -(C_{0}N-c)\parallel u_t{\parallel }_2^2-{\beta }_1{\parallel \nabla u\parallel }_2^2-(C_{0}N-c){\parallel z(x,1,t)\parallel }_2^2\hfill \\ \hfill & -N{\mathfrak{B}}_1{\int }_{{\Sigma }_0}{\int }_{-\infty }^{+\infty }({\xi }^2+\eta )|\varphi (\xi ,t){|}^{2}d\xi d\Sigma -(N{p}_0-c){\parallel {\chi }_t\parallel }_{{\Sigma }_0}^2\hfill \\ \hfill & -\mathfrak{M}E\left(t\right)-{\beta }_2{\int }_{{\Sigma }_0}{\int }_{-\infty }^{+\infty }{\left|\varphi (\xi ,t)\right|}^{2}d\xi d\Sigma -{\beta }_3{\int }_0^1{\parallel z(x,\varpi ,t)\parallel }_2^{2}d\varpi .\hfill \end{array}$$

(57)

However, applying Poincaré’s and Young’s inequalities, we obtain

$$|{\Psi }_1\left(t\right)|\le {\displaystyle \frac{1}{2}}\parallel u_t{\parallel }_2^2+{\displaystyle \frac{\widehat{B}}{2}}{\parallel \nabla u\parallel }_2^2+{\displaystyle \frac{1+p_1}{2}}{\parallel \chi \parallel }_{{\Sigma }_0}^2,$$

(58)

• where $\widehat{B}=(1+{\mathfrak{A}}_1)B_{1,\Omega }^2+B_{1,{\Sigma }_0}^2$.
• Also, from (41), we find

$${{\rm Y}}^2(x,\xi ,t)={\left|\varphi (\xi ,t)\right|}^2+{\left|u(x,t)\right|}^2{\mu }^2\left(\xi \right)-2\varphi (\xi ,t)u(x,t)\mu \left(\xi \right).$$

(59)

After dividing the last equality by $({\xi }^2+\eta )$ and integrating the result over ${\Sigma }_0\times (-\infty ,+\infty )$, we find

$$\begin{array}{ccc}\hfill |{\Psi }_2\left(t\right)|& \le & {\int }_{{\Sigma }_0}{\int }_{-\infty }^{+\infty }{\displaystyle \frac{{\left|\varphi (\xi ,t)\right|}^2}{({\xi }^2+\eta )}}d\xi d\Sigma +{\int }_{{\Sigma }_0}{\int }_{-\infty }^{+\infty }{\displaystyle \frac{{\left|u(x,t)\right|}^2{\mu }^2\left(\xi \right)}{({\xi }^2+\eta )}}d\xi d\Sigma \hfill \\ & & +2{\int }_{{\Sigma }_0}{\int }_{-\infty }^{+\infty }{\displaystyle \frac{\left|\varphi \right(\xi ,t\left)u\right(x,t\left)\mu \right(\xi \left)\right|}{({\xi }^2+\eta )}}d\xi d\Sigma .\hfill \end{array}$$

(60)

Utilizing Young’s inequality, we obtain $\forall \delta >0$:

$$\begin{array}{cc}\hfill {\int }_{{\Sigma }_0}{\int }_{-\infty }^{+\infty }{\displaystyle \frac{\left|\varphi \right(\xi ,t\left)u\right(x,t\left)\mu \right(\xi \left)\right|}{({\xi }^2+\eta )}}d\xi d\Sigma & \le {\displaystyle \frac{1}{4\delta }}{\int }_{{\Sigma }_0}{\int }_{-\infty }^{+\infty }{\displaystyle \frac{{\left|\varphi (\xi ,t)\right|}^2}{({\xi }^2+\eta )}}d\xi d\Sigma \hfill \\ \hfill & +\delta {\int }_{{\Sigma }_0}{\int }_{-\infty }^{+\infty }{\displaystyle \frac{{\left|u(x,t)\right|}^2{\mu }^2\left(\xi \right)}{({\xi }^2+\eta )}}d\xi d\Sigma .\hfill \end{array}$$

(61)

Combining (60) and (61), and using the estimate ${\displaystyle \frac{1}{{\xi }^2+\eta }}<{\displaystyle \frac{1}{\eta }}$, we obtain

$$\begin{array}{ccc}\hfill |{\Psi }_2\left(t\right)|& \le & \left({\displaystyle \frac{2\delta +1}{2\delta \eta }}\right){\int }_{{\Sigma }_0}{\int }_{-\infty }^{+\infty }{\left|\varphi (\xi ,t)\right|}^{2}d\xi d\Sigma \hfill \\ & & +{\displaystyle \frac{(2\delta +1)}{\eta }}{\int }_{{\Sigma }_0}{\left|u(x,t)\right|}^2{\int }_{-\infty }^{+\infty }{\mu }^2\left(\xi \right)d\xi d\Sigma .\hfill \end{array}$$

(62)

Lemma 2 and Lemma 2.1 in [20] gives

$$|{\Psi }_2\left(t\right)|\le \left({\displaystyle \frac{2\delta +1}{2\delta \eta }}\right){\int }_{{\Sigma }_0}{\int }_{-\infty }^{+\infty }|\varphi (\xi ,t){|}^{2}d\xi d\Sigma +A_0{B}_{1,{\Sigma }_0}^2{\displaystyle \frac{(2\delta +1)}{\eta }}{\parallel \nabla u\parallel }_2^2.$$

(63)

Next, we have

$$|{\Psi }_3\left(t\right)|\le c_1{\int }_{\Omega }{\int }_0^1{z}^2\left(x,\varpi ,t\right)d\varpi dx.$$

(64)

Adding (58) and (63) to (64), we obtain

$$\begin{array}{ccc}\hfill |{\Psi }_1\left(t\right)+{\displaystyle \frac{{\mathfrak{B}}_1}{2}}{\Psi }_2\left(t\right)+{\Psi }_3\left(t\right)|& \le & |{\Psi }_1\left(t\right)|+{\displaystyle \frac{{\mathfrak{B}}_1}{2}}|{\Psi }_2\left(t\right)|+|{\Psi }_3\left(t\right)|\hfill \\ & \le & {\displaystyle \frac{1}{2}}\parallel u_t{\parallel }_2^2+c_1{\int }_{\Omega }{\int }_0^1{z}^2\left(x,\varpi ,t\right)d\varpi dx+{\displaystyle \frac{1+p_1}{2}}{\parallel \chi \parallel }_{{\Sigma }_0}^2\hfill \\ & & +{\displaystyle \frac{1}{2}}\left[A_0{B}_{1,{\Sigma }_0}^2{\mathfrak{B}}_1{\displaystyle \frac{(2\delta +1)}{\eta }}+\widehat{B}\right]{\parallel \nabla u\parallel }_2^2\hfill \\ & & +{\displaystyle \frac{{\mathfrak{B}}_1}{2}}\left[{\displaystyle \frac{2\delta +1}{2\delta \eta }}\right]{\int }_{{\Sigma }_0}{\int }_{-\infty }^{+\infty }{\left|\varphi (\xi ,t)\right|}^{2}d\xi d\Sigma .\hfill \end{array}$$

(65)

Hence, by the definition of energy (14), for $\gamma >0$, we find

$$\begin{array}{cc}\hfill \left|\mathcal{L}\right(t)-NE(t\left)\right|& \le \gamma E\left(t\right)+{\displaystyle \frac{1-\gamma }{2}}\parallel u_t{\parallel }_2^2+{\displaystyle \frac{\gamma }{p}}{\int }_{\Omega }ln\left|u\right|u^{p}dx-{\displaystyle \frac{\gamma }{p^2}}{\parallel u\parallel }_p^p\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}\left[A_0{B}_{1,{\Sigma }_0}^2{\mathfrak{B}}_1{\displaystyle \frac{(2\delta +1)}{\eta }}+\widehat{B}-\gamma \right]{\parallel \nabla u\parallel }_2^2\hfill \\ \hfill & +{\displaystyle \frac{{\mathfrak{B}}_1}{2}}\left[{\displaystyle \frac{2\delta +1}{2\delta \eta }}-\gamma \right]{\int }_{{\Sigma }_0}{\int }_{-\infty }^{+\infty }{\left|\varphi (\xi ,t)\right|}^{2}d\xi d\Sigma \hfill \\ \hfill & +(c_1-{\displaystyle \frac{\gamma \Theta }{2}}){\int }_{\Omega }{\int }_0^1{z}^2\left(x,\varpi ,t\right)d\varpi dx\hfill \\ \hfill & +({\displaystyle \frac{1+p_1-\gamma q_0}{2}}){\parallel \chi \parallel }_{{\Sigma }_0}^2.\hfill \end{array}$$

Finally, by (52), we obtain

$$\begin{array}{ccc}\hfill \left|\mathcal{L}\right(t)-NE(t\left)\right|& \le & \gamma E\left(t\right)+{\displaystyle \frac{1-\gamma }{2}}{\parallel u_t\parallel }_2^2+(c_1-{\displaystyle \frac{\gamma \Theta }{2}}){\int }_{\Omega }{\int }_0^1{z}^2\left(x,\varpi ,t\right)d\varpi dx\hfill \\ & & +{\displaystyle \frac{1}{2}}\left[A_0{B}_{1,{\Sigma }_0}^2{\mathfrak{B}}_1{\displaystyle \frac{(2\delta +1)}{\eta }}+\widehat{B}-\gamma {\displaystyle \frac{p-2}{p}}\right]{\parallel \nabla u\parallel }_2^2\hfill \\ & & +{\displaystyle \frac{{\mathfrak{B}}_1}{2}}\left[{\displaystyle \frac{2\delta +1}{2\delta \eta }}-\gamma \right]{\int }_{{\Sigma }_0}{\int }_{-\infty }^{+\infty }{\left|\varphi (\xi ,t)\right|}^{2}d\xi d\Sigma \hfill \\ & & +({\displaystyle \frac{1+p_1-\gamma q_0}{2}}){\parallel \chi \parallel }_{{\Sigma }_0}^2,\hfill \end{array}$$

(66)

by picking $\gamma$ and $\delta$ such that

$$\gamma >max\left\{1,{\displaystyle \frac{2\delta +1}{2\delta \eta }},{\displaystyle \frac{p(A_0{B}_{1,{\Sigma }_0}^2{\mathfrak{B}}_1(2\delta +1)+\widehat{B}\eta )}{\eta (p-2)}},{\displaystyle \frac{2c_1}{\Theta }},{\displaystyle \frac{1+p_1}{q_0}}\right\}.$$

Then, from (66), we find

$$(N-\gamma )E\left(t\right)\le \mathcal{L}\left(t\right)\le (N+\gamma )E\left(t\right).$$

(67)

Now, we make N so large that

$$C_{0}N-c>0,N-\gamma >0,N{p}_0-\gamma >0.$$

From (57) and (67), we obtain for some ${\lambda }_1,{\lambda }_2>0$:

$$\left\{\begin{array}{c}{\lambda }_{1}E\left(t\right)\le \mathcal{L}\left(t\right)\le {\lambda }_{2}E\left(t\right),\hfill \\ {\mathcal{L}}^{\prime }\left(t\right)\le -\mathfrak{M}E\left(t\right)\le {\displaystyle \frac{-\mathfrak{M}}{{\lambda }_2}}\mathcal{L}\left(t\right).\hfill \end{array}\right.$$

(68)

A direct calculation of the integrate of (68) over $(0,t)$ gives

$$\mathcal{L}\left(t\right)\le \mathcal{L}\left(0\right)e^{-k_{1}t},\forall t\ge 0.$$

(69)

An amalgamation of (68) and (69) gives (47). This concludes the proof. □

5. Conclusions

This paper examined the global existence and general decay of solutions to a wave equation characterized by coupling with logarithmic source and delay terms and governed by both fractional and acoustic boundary conditions. The global existence of solutions was analyzed under a range of hypotheses, and the general decay behavior was established through the construction and application of an appropriate Lyapunov function. This study provides valuable insights into the global existence and asymptotic behavior of solutions to a nonlinear wave equation with acoustic and fractional boundary conditions, coupled by logarithmic source and delay terms. In future work, we will focus on numerical methods to validate and extend the analytic findings of this work.