On Construction of Solutions of Hyperbolic Kirchhoff-Type Problems Involving Free Boundary and Volume Constraint

Abstract

In this paper, we have undertaken the challenging and novel task of establishing the existence of weak solutions for four types of hyperbolic Kirchhoff-type problems: the classical hyperbolic Kirchhoff problem, the problem with a free boundary, the problem with a volume constraint, and the problem combining both a volume constraint and a free boundary. These problems are characterized by the presence of non-local terms arising from the Kirchhoff term, the free boundary, and the volume constraint, which introduces significant analytical complexities. To address these challenges, we utilize the discrete Morse flow (DMF) approach, reformulating the original continuous problem into a sequence of discrete minimization problems. This method guarantees the existence of a minimizer for the discretized functional, which subsequently serves as a weak solution to the primary problem.

1. Introduction

We address in this paper the existence of weak solutions for the hyperbolic Kirchhoff-type problems involving a free boundary and a volume constraint by studying the following models:

$$\left\{\begin{array}{c}\begin{array}{cc}{v}_{tt}(x,t)=\left(a+b{∥\nabla v(x,t)∥}_{{\mathrm{L}}^2(\Omega )}^2\right)\Delta v(x,t)-f(x,t,v),\hfill & \mathrm{in}{\Omega }_T,\hfill \\ v(x,t)=0,\hfill & \mathrm{on}(0,T)\times \partial \Omega ,\hfill \\ v(x,0)=v_0\left(x\right),\hfill & \mathrm{in}\Omega ,\hfill \\ v_t(x,0)=u_0\left(x\right),\hfill & \mathrm{in}\Omega ,\hfill \end{array}\hfill \end{array}\right.$$

(1)

$$\left\{\begin{array}{c}\begin{array}{cc}{\chi }_{\left\{v>0\right\}}{v}_{tt}(x,t)=\left(a+b{∥\nabla v(x,t)∥}_{{\mathrm{L}}^2(\Omega )}^2\right)\Delta v(x,t)-f(x,t,v),\hfill & \mathrm{in}{\Omega }_T,\hfill \\ v(x,t)=0,\hfill & \mathrm{on}(0,T)\times \partial \Omega ,\hfill \\ v(x,0)=v_0\left(x\right),\hfill & \mathrm{in}\Omega ,\hfill \\ v_t(x,0)=u_0\left(x\right),\hfill & \mathrm{in}\Omega ,\hfill \end{array}\hfill \end{array}\right.$$

(2)

$$\left\{\begin{array}{c}\begin{array}{cc}{v}_{tt}(x,t)=\left(a+b{∥\nabla v(x,t)∥}_{{\mathrm{L}}^2(\Omega )}^2\right)\Delta v(x,t)-f(x,t,v)+\lambda \left(v\right),\hfill & \mathrm{in}{\Omega }_T,\hfill \\ v(x,t)=0,\hfill & \mathrm{on}(0,T)\times \partial \Omega ,\hfill \\ v(x,0)=v_0\left(x\right),\hfill & \mathrm{in}\Omega ,\hfill \\ v_t(x,0)=u_0\left(x\right),\hfill & \mathrm{in}\Omega ,\hfill \end{array}\hfill \end{array}\right.$$

(3)

$$\left\{\begin{array}{c}\begin{array}{cc}{\chi }_{\left\{v>0\right\}}{v}_{tt}(x,t)=\left(a+b{∥\nabla v(x,t)∥}_{{\mathrm{L}}^2(\Omega )}^2\right)\Delta v(x,t)-f(x,t,v)+{\chi }_{\left\{v>0\right\}}\lambda \left(v\right),\hfill & \mathrm{in}{\Omega }_T,\hfill \\ v(x,t)=0,\hfill & \mathrm{on}(0,T)\times \partial \Omega ,\hfill \\ v(x,0)=v_0\left(x\right),\hfill & \mathrm{in}\Omega ,\hfill \\ v_t(x,0)=u_0\left(x\right),\hfill & \mathrm{in}\Omega ,\hfill \end{array}\hfill \end{array}\right.$$

(4)

where $\Omega \subset {\mathbb{R}}^N$ is a bounded domain with a Lipschitz boundary, and ${\Omega }_T=\Omega \times (0,T)$ with time $T>0$, $v:{\Omega }_T\to \mathbb{R}$ is a scalar function.

• ${\chi }_{\left\{v>0\right\}}$ refers to the following set’s characteristic function:

  $$\left\{v>0\right\}:=\left\{(x,t)\in {\Omega }_T;v(x,t)>0\right\}.$$

  (5)
• $f:{\Omega }_T\times \mathbb{R}\to \mathbb{R}$ is a function satisfies that:
  • f is a Carathéodory function.
  • There is a constant $C_f$ that satisfies

    $${\displaystyle 0<C_f<{\frac{a}{4}}^{\frac{1}{2}},}$$

    (6)

    and a function $\delta \in L^2\left({\Omega }_T\right)$, such that

    $$\left|f(x,t,v)\right|\le C_f\left|v(x,t)\right|+\delta (x,t),\mathrm{for}\mathrm{a}.\mathrm{e}.(x,t)\in {\Omega }_T,$$

    (7)
• $\lambda$ represents a non-local term arising from the volume constraint applied on the unknown function v.

  $${\int }_{\Omega }v(x,t)dx=V,\forall t\in [0,T],$$

  (8)

  with $V>0$ representing the volume, and the non-local term $\lambda$ is given by

  $$\lambda \left(v\right)={\displaystyle \frac{1}{V}}{\int }_{\Omega }\left(v_{tt}(x,t)-\left(a+b{∥\nabla v(x,t)∥}_{{\mathrm{L}}^2(\Omega )}^2\right)\Delta v(x,t)+f(x,t,v)\right)v(x,t)dx.$$

  (9)

  We note that if $\lambda$ does not vanish, its presence guarantees that the solutions of (1) adhere to the volume constraint (8).

The problem (1) represents a model for the free vibration of an elastic string. The string’s displacement is described by the function $v(x,t)$, while the function $f(x,t,v)$ accounts for any external forces acting on the string. The parameters a and b characterize the string’s material properties and the initial tension, respectively. The governing wave equation takes into account factors such as the string’s tension, mass per unit length, and the initial displacement and velocity. Additionally, variations in the string’s length due to its transverse vibrations are considered. The Dirichlet boundary condition enforces the string’s fixed endpoints, meaning the displacement is zero at both ends, implying that the string is rigidly anchored at these points. By imposing this boundary condition, we can effectively describe the string’s vibrational behavior, determine its natural frequencies, and examine the modes of vibration and the corresponding waveforms. The free vibration of elastic strings has numerous real-life applications across various fields. In musical instruments like guitars, violins, and pianos, the vibration of strings is modeled to achieve specific tones and frequencies. In engineering, the dynamics of cables in suspension bridges, power lines, and tensioned ropes are analyzed to ensure structural stability and safety. Similarly, seismic wave propagation in the Earth’s crust during earthquakes is modeled using these principles to aid in prediction and mitigation. In telecommunications, the behavior of waves in fiber optics and waveguides is studied to improve signal transmission. Additionally, this phenomenon is crucial for optimizing sports equipment, such as tennis racket strings and bowstrings, and is applied in biomedical imaging, where the vibration of biological membranes and tissues is analyzed for technologies like ultrasound.

The problem (2) presents a model for the free vibration of an elastic string on a flat surface. The string’s displacement is described by the function v, and $f(x,t,v)$ represents the external force acting on the string. The parameters a and b denote the intrinsic properties of the string and the initial tension, respectively. The displacement function $v(x,t)$ is constrained to lie on the flat surface, so $v(x,t)\ge 0$ remains valid. The wave equation incorporates various factors influencing the behavior of the vibrating string, such as tension, mass per unit length, and initial conditions, including the initial displacement and velocity of the string. Kirchhoff’s model further accounts for variations in the length of the string due to transverse vibrations. The Dirichlet boundary condition specifies a string with fixed endpoints at equilibrium, meaning the string is anchored at these points and cannot move. By applying the Dirichlet boundary condition with fixed endpoints, we can effectively describe the string’s behavior and determine its vibrational modes, frequencies, and corresponding waveforms. The free vibration of elastic strings constrained on flat surfaces has specialized applications. In percussion instruments like pianos, the interaction of strings with rigid surfaces influences their vibrational behavior and sound production. In mechanical systems, it applies to tensioned cables or strings interacting with flat surfaces, such as those found in conveyor belts and guided machinery. Structural engineering uses these models to study scenarios where wires or cables contact rigid boundaries, like bridge supports. In biomedical devices, the constrained vibration of membranes or tissues is crucial for diagnostic and therapeutic applications. Additionally, robotic systems utilize such models for cable-driven actuators, and material science benefits from studying the dynamics of constrained wires or fibers for stability and design improvements.

The problem (3) is a model for the free vibration of an elastic membrane covering a container filled with an incompressible liquid. The membrane’s displacement is described by the function v, and the function $f(x,t,v)$ is presented as an external force. The parameters a and b denote the intrinsic properties of the membrane and the initial tension, respectively. Moreover, the volume of the liquid inside the membrane does not change. The wave equation considers various factors that influence the behavior of the vibrating membrane, including tension, mass per unit length, and initial conditions such as the initial displacement and velocity of the membrane. Furthermore, Kirchhoff’s model considers the variations in the length of the membrane caused by transverse vibrations. The Dirichlet boundary condition specifies a membrane with fixed endpoints at equilibrium, meaning that the membrane is anchored or fixed at those points and cannot move. By imposing the Dirichlet boundary condition with fixed endpoints, we can accurately describe the membrane’s behavior and determine its vibrational modes, frequencies, and corresponding waveforms. This phenomenon has various applications in engineering and biomedical fields. It is critical in the design and analysis of flexible storage tanks or membranes used in fuel tanks, water reservoirs, and industrial liquid containers, where the membrane’s vibration affects stability. In biomechanics, it models the behavior of biological membranes such as the diaphragm in the human body, which interacts with incompressible fluids like blood or organ fluids. Aerospace engineering employs this principle in fuel tanks of spacecraft, where the flexible membranes interact with liquids to stabilize sloshing during motion. Additionally, underwater acoustics uses these models to study how elastic membranes interact with incompressible fluids in sonar systems and underwater devices.

The problem (4) is a model for the free vibration of an elastic membrane covering a container filled with an incompressible liquid on a flat surface. The membrane’s displacement is described by the function v, and $f(x,t,v)$ is presented as the external force. The parameters a and b denote the intrinsic properties of the membrane and the initial tension, respectively. The displacement function $v(x,t)$ is restricted to the surface, so then $v(x,t)\ge 0$ is restricted to the surface. Moreover, the volume of the liquid inside the membrane does not change. The wave equation considers various factors that influence the behavior of the vibrating membrane, including tension, mass per unit length, and initial conditions such as the initial displacement and velocity of the membrane. Furthermore, Kirchhoff’s model considers the variations in the length of the membrane caused by transverse vibrations. The Dirichlet boundary condition specifies a membrane with fixed endpoints at equilibrium, meaning that the membrane is anchored or fixed at those points and cannot move. By imposing the Dirichlet boundary condition with fixed endpoints, we can accurately describe the membrane’s behavior and determine its vibrational modes, frequencies, and corresponding waveforms. This variation is specifically relevant where the container is constrained by a flat surface, leading to distinct vibrational behaviors. Applications include liquid storage systems where the container’s base is rigidly supported, such as tanks used in transportation vehicles or industrial platforms. In medical devices, this principle is applied in analyzing vibrations of elastic membranes in contact with flat surfaces, like flexible diaphragms in pumps or drug delivery systems. Robotics utilizes this model to optimize liquid-carrying systems with flexible membranes supported on rigid platforms. Additionally, it is important in fluid-structure interaction studies, such as analyzing the damping effects of flat surfaces on vibrational stability in engineering designs.

The objective of this article is to prove the existence of weak solutions by using a variational approach. The variational approach that we used to achieve this goal is known as the discrete Morse flow (DMF) of hyperbolic type. The DMF is described as a variational approach employed for time-dependent problems. It involves transforming the main problem into a series of minimization problems defined at discrete time intervals. The discretized functional is stated to be non-negative, which allows for the demonstration of the existence of a minimizer, offering a significant benefit compared to the functional itself. The minimizer of the functional serves as a solution to the discretized problem, providing the main problem with a weak solution (interested readers see [1]). This variational approach was first introduced in [2] with N. Kikuchi. It was first used on parabolic type problems, and then developed for the hyperbolic type as in [3] by K. Hoshino and N. Kikuchi.The DMF has been successfully used to solve various types of problems, including free boundary problems, volume-preserving problems, and problems involving non-local terms. In [4] by E. Ginder and K. Švadlenka, they prove the existence of a weak solution to a free boundary hyperbolic-type problem with a volume constraint using the DMF. This versatility makes it an ideal choice for addressing problems similar to the one at hand. We will be able to show the existence of a weak solution as in [5] by K. Kikuchi and [6] by Y. Akagawa, who present a free boundary hyperbolic problem involving modeling the free vibrations of elastic string similar to ours, but without the Kirchhoff term, the external force term, and the volume constraint. They show the existence of a weak solution in one dimension. Furthermore, Y. Akagawa developed a scheme that guarantees energy preservation, and the difference between both schemes is given. Moreover, our approximation approach’s energy-preserving properties could be used not only to create robust numerical algorithms but also to prove the uniqueness of solutions for particular wave-type problems. Additionally, ref. [7] by M. Bonafini considers a non-local wave equation obstacle problem and demonstrates the existence of a distinct (weaker) version of a weak solution using a convex minimization approach utilizing a time-discrete approximation scheme. Furthermore, an important aspect of our approach is that a uniform estimate is given for the approximate solutions, allowing the identification of a unique limit function. Another interesting approach presented in [8] by G. Dal Maso is to recast a nonlinear wave-type development as a minimization problem. Instead of discretizing time, this technique exploits a singular weight in the integrand, which neatly leads to a convex minimization problem and proves the existence of weak solutions. It is that a uniform estimate is given for the approximate solutions, allowing the identification of a unique limit function. The fact that the approximate solution applies to any test function compactly supported in limit function support is a result of the uniform convergence of the approximation sequence, and that is an advantage to our approach, unlike the one given by G. Dal Maso.

In this paper, drawing inspiration from previous work and [9], we propose a method to prove the existence of weak solutions for a new hyperbolic Kirchhoff-type problem involving a free boundary and volume constraint, using the DMF approach. This manuscript not only builds upon the foundational results presented in [9] but also incorporates them to provide a clearer and more professional presentation. Additionally, we extend the analysis to address other types of Kirchhoff problems, including the classical problem and those combining free boundaries and volume constraints, thereby broadening the scope and impact of this research.

We begin by defining the weak solution of problem (1) and (3) and providing the rationale for it. We set the test functions $\varphi (x,t)$ to the space $C_0^{\infty }(\Omega \times [0,T))$. Therefore, similar to [1], we have the following definition.

Definition 1.

The function $v\in {\mathrm{H}}^1(0,T;{\mathrm{L}}^2(\Omega ))\cap {\mathrm{L}}^{\infty }(0,T;{\mathrm{H}}_0^1(\Omega ))$ is defined to be the weak solution for problem (1) if for all $\varphi (x,t)\in C_0^{\infty }((\Omega \times [0,T))$, and $v(x,0)=v_0$, the following holds:

$$\begin{array}{cc}\hfill {\int }_0^T& {\int }_{\Omega }\left[-v_t(x,t){\varphi }_t(x,t)+\left(a+b{∥\nabla v(x,t)∥}_{{\mathrm{L}}^2(\Omega )}^2\right)\nabla v(x,t)\nabla \varphi (x,t)+f(x,t,v)\varphi (x,t)\right]dxdt\hfill \\ & -{\int }_{\Omega }{u}_0\left(x\right)\varphi (x,0)dx=0.\hfill \end{array}$$

(10)

Definition 2.

The function $v\in {\mathrm{H}}^1(0,T;{\mathrm{L}}^2(\Omega ))\cap {\mathrm{L}}^{\infty }(0,T;{\mathrm{H}}_0^1(\Omega ))$ is defined to be the weak solution for problem (1) if for all $\varphi (x,t)\in C_0^{\infty }((\Omega \times [0,T))$, and $v(x,0)=v_0$, the following holds:

$$\begin{array}{cc}\hfill {\int }_0^T& {\int }_{\Omega }\left[-v_t(x,t){\varphi }_t(x,t)+\left(a+b{∥\nabla v(x,t)∥}_{{\mathrm{L}}^2(\Omega )}^2\right)\nabla v(x,t)\nabla \varphi (x,t)+f(x,t,v)\varphi (x,t)\right]dxdt\hfill \\ & -{\int }_{\Omega }{u}_0\left(x\right)\varphi (x,0)dx=-{\displaystyle \frac{1}{V}}{\int }_{\Omega }{u}_0\left(x\right)v_0\left(x\right)\Phi \left(0\right)dx-{\displaystyle \frac{1}{V}}{\int }_0^T{\int }_{\Omega }{v}_t(x,t){(v\Phi )}_t(x,t)dxdt\hfill \\ & +{\displaystyle \frac{1}{V}}{\int }_0^T{\int }_{\Omega }\left(\left(a+b{∥\nabla v(x,t)∥}_{{\mathrm{L}}^2(\Omega )}^2\right){\left|\nabla v(x,t)\right|}^2+f(x,t,v)v(x,t)\right)\Phi \left(t\right)dxdt,\hfill \end{array}$$

(11)

where

$$\Phi \left(t\right)={\int }_{\Omega }\varphi (x,t)dx.$$

(12)

For the problems (2) and (4), in defining the weak solution and providing the rationale for it, the free boundary regularity will be required to obtain this weak solution, and it is unclear whether a solution of this kind exists because we cannot manage the free boundary regularity. We must limit the test functions to the space $C_0^{\infty }((\Omega \times [0,T))\cap \left(\left\{v>0\right\}\right))$. We see that examining this kind of set of test functions makes sense if and only if v is sufficiently regular. In this case, continuity is enough to ensure that $\left\{v>0\right\}$ is an open set. Nevertheless, we can obtain another weak solution definition that, if one can show the free boundary regularity, coincides with Definitions 1 and 2.

Definition 3.

The function $v\in {\mathrm{H}}^1(0,T;{\mathrm{L}}^2(\Omega ))\cap {\mathrm{L}}^{\infty }(0,T;{\mathrm{H}}_0^1(\Omega ))$ is defined to be the weak solution for problem (2) if $v=0$ outgoing $\left\{v>0\right\}$,$v(x,0)=v_0$, and for all $\varphi (x,t)\in C_0^{\infty }((\Omega \times [0,T))\cap \left\{v>0\right\})$, the following holds:

$$\begin{array}{cc}\hfill & {\int }_0^T{\int }_{\Omega }\left[-v_t(x,t){\varphi }_t(x,t)+\left(a+b{∥\nabla v(x,t)∥}_{{\mathrm{L}}^2(\Omega )}^2\right)\nabla v(x,t)\nabla \varphi (x,t)\right]dxdt\hfill \\ & +{\int }_0^T{\int }_{\Omega }f(x,t,v)\varphi (x,t)dxdt-{\int }_{\Omega }{v}_0\left(x\right)\varphi (x,0)dx=0.\hfill \end{array}$$

(13)

Definition 4.

The function $v\in {\mathrm{H}}^1(0,T;{\mathrm{L}}^2(\Omega ))\cap {\mathrm{L}}^{\infty }(0,T;{\mathrm{H}}_0^1(\Omega ))$ is defined to be the weak solution for problem (2) if $v=0$ outgoing $\left\{v>0\right\}$,$v(x,0)=v_0$, and for all $\varphi (x,t)\in C_0^{\infty }((\Omega \times [0,T))\cap \left\{v>0\right\})$, the following holds:

$$\begin{array}{cc}\hfill & {\int }_0^T{\int }_{\Omega }\left[-v_t(x,t){\varphi }_t(x,t)+\left(a+b{∥\nabla v(x,t)∥}_{{\mathrm{L}}^2(\Omega )}^2\right)\nabla v(x,t)\nabla \varphi (x,t)\right]dxdt\hfill \\ & +{\int }_0^T{\int }_{\Omega }f(x,t,v)\varphi (x,t)dxdt-{\int }_{\Omega }{v}_0\left(x\right)\varphi (x,0)dx=-{\displaystyle \frac{1}{V}}{\int }_{\Omega }\tilde{v}(x,0)v_0\left(x\right)\Phi \left(0\right)dx\hfill \\ & -{\displaystyle \frac{1}{V}}{\int }_0^T{\int }_{\Omega }{v}_t(x,t){(\tilde{v}\Phi )}_t(x,t)dxdt+{\displaystyle \frac{1}{V}}{\int }_0^T{\int }_{\Omega }f(x,t,v)\tilde{v}(x,t)\Phi \left(t\right)dxdt\hfill \\ & +{\displaystyle \frac{1}{V}}{\int }_0^T{\int }_{\Omega }\left(a+b{∥\nabla v(x,t)∥}_{{\mathrm{L}}^2(\Omega )}^2\right)\nabla v(x,t)\nabla \tilde{v}(x,t)\Phi \left(t\right)dxdt,\hfill \end{array}$$

(14)

for $\tilde{v}\in C_0^{\infty }(([0,T]\times \Omega )\cap \{v>0\})$ an arbitrary function satisfies ${\displaystyle {\int }_{\Omega }\tilde{v}\mathrm{d}x=V}$, with

$$\Phi \left(t\right)={\int }_{\Omega }\varphi (x,t)dx.$$

(15)

The primary result of this work is stated in the theorem that follows.

Theorem 1.

For $u_0$ belongomg to ${\mathrm{H}}_0^1(\Omega )$, and $v_0$ in ${\mathrm{H}}_0^1(\Omega )$, then the problems (1)–(4) have a weak solution.

In the remainder of this paper, our objective is to prove the existence of the weak solutions defined in Definitions 1–4. We begin by constructing the approximate weak solution in Section 2. This involves first introducing the functionals ${\mathcal{F}}^i,i=1,\dots ,4$, then subsequently demonstrating the existence of a minimizer, which leads to the derivation of the approximate weak solution. In Section 4, we will focus on refining the time step as k tends to zero and examining the convergence process. Throughout this analysis, we will employ the limit passage method to validate the weak solution’s existence. An essential component of this investigation is the energy estimate, which will be thoroughly discussed in Section 3.

2. Construction of the Approximate Weak Solution

In this work, we use a minimizing movement known as the DMF approach to solve our problem. This approach depends on the functional discrete-time level minimization. The DMF approach is unique because it may be utilized to obtain both theoretical and numerical findings. Moreover, the discretization lets us employ elliptic theory results. We recommended [1] for interested readers and further information on this strategy.

We start by partitioning the time interval $(0,T)$ into M equal sub-intervals and take $k=T/M$. Using our initial conditions, and by the use of the backward difference, we define $v_{-1}$ as follows:

$$v_{-1}=v_0-k{u}_0.$$

(16)

Then, we identify $v_m\in {\mathrm{H}}_0^1(\Omega )$ with $m=1,\dots ,M$, to minimize the functional

$${\mathcal{F}}_m^1\left(v\right)={\int }_{\Omega }\left[{\displaystyle \frac{{\left|v-2v_{m-1}+v_{m-2}\right|}^2}{2k^2}}+\left({\displaystyle \frac{a}{2}}+{\displaystyle \frac{b}{4}}{∥\nabla v∥}_{{\mathrm{L}}^2(\Omega )}^2\right){\left|\nabla v\right|}^2+F_m\left(v\right)\right]dx,$$

(17)

$${\mathcal{F}}_m^2\left(v\right)={\int }_{\Omega }\left[{\displaystyle \frac{{\left|v-2v_{m-1}+v_{m-2}\right|}^2}{2k^2}}+\left({\displaystyle \frac{a}{2}}+{\displaystyle \frac{b}{4}}{∥\nabla v∥}_{{\mathrm{L}}^2(\Omega )}^2\right){\left|\nabla v\right|}^2+F_m\left(v\right)\right]dx+{\Psi }^1\left(v\right),$$

(18)

$${\mathcal{F}}_m^3\left(v\right)={\int }_{\Omega }\left[{\displaystyle \frac{{\left|v-2v_{m-1}+v_{m-2}\right|}^2}{2k^2}}+\left({\displaystyle \frac{a}{2}}+{\displaystyle \frac{b}{4}}{∥\nabla v∥}_{{\mathrm{L}}^2(\Omega )}^2\right){\left|\nabla v\right|}^2+F_m\left(v\right)\right]dx+{\Psi }^2\left(v\right),$$

(19)

$${\mathcal{F}}_m^4\left(v\right)={\int }_{\Omega }\left[{\displaystyle \frac{{\left|v-2v_{m-1}+v_{m-2}\right|}^2}{2k^2}}+\left({\displaystyle \frac{a}{2}}+{\displaystyle \frac{b}{4}}{∥\nabla v∥}_{{\mathrm{L}}^2(\Omega )}^2\right){\left|\nabla v\right|}^2+F_m\left(v\right)\right]dx+{\Psi }^1\left(v\right)+{\Psi }^2\left(v\right),$$

(20)

where

$$\begin{array}{cc}{\Psi }^1\left(v\right)=\left\{\begin{array}{c}{\Psi }^1=0;\mathrm{if}v(x,t)\ge 0,\hfill \\ {\Psi }^1=\infty ;\mathrm{otherwise},\hfill \end{array}\right.\hfill & \mathrm{and}{\Psi }^2\left(v\right)=\left\{\begin{array}{c}{\displaystyle {\Psi }^2=0;\mathrm{if}{\int }_{\Omega }v(x,t)dx=V,}\hfill \\ {\Psi }^2=\infty ;\mathrm{otherwise}.\hfill \end{array}\right.\hfill \end{array}$$

(21)

We discretized the term f at time level $m=1,2,\dots ,M$ by means of introducing

$$F_m(x,v)={\int }_0^v{f}_m(x,s)ds,\mathrm{where}{f}_m(x,v)={\displaystyle \frac{1}{k}}{\int }_{(m-1)k}^{mk}f(x,t,v)dt.$$

(22)

Furthermore,

$$\left|F_m\left(v\right)\right|\le {\displaystyle \frac{C_f}{2}}{\left|v\right|}^2+{\delta }_m\left(x\right)\left|v\right|,\mathrm{where}{\delta }_m\left(x\right)={\displaystyle \frac{1}{k}}{\int }_{(m-1)k}^{mk}\delta (x,t)dt.$$

(23)

In addition, we suppose that the initial data $u_0$ belong to ${\mathrm{H}}_0^1(\Omega )$, and $v_0$ in ${\mathrm{H}}_0^1(\Omega )$. These constraints will be required to assure the existence of functional minimizers of the functional ${\mathcal{F}}_m^1$, as will be seen later.

Taking $v_{-1}$ and $v_0$ belonging to ${\mathrm{H}}_0^1(\Omega )$, we define $v_1$ as a minimizer of ${\mathcal{F}}_1^1$ and ${\mathcal{F}}_1^2$. Therefore, $v_m$ is defined iteratively to be a minimizer of ${\mathcal{F}}_m^i,i=1,\dots ,4$. In general, the concept of the DMF is to utilize the minimizers as mentioned earlier to generate an approximate solution via temporal interpolation and then take k approach zero to achieve the weak solution that we are looking for. As regards ${\mathcal{F}}_m^i,i=1,\dots ,4$, we can see that ${\Psi }^1$ is an indicator function that ensures that each minimizer of ${\mathcal{F}}_m^2$ and ${\mathcal{F}}_m^3$ is positive, and ${\Psi }^2$ is an indicator function that ensures that each minimizer of ${\mathcal{F}}_m^3$ and ${\mathcal{F}}_m^4$ satisfies the volume constraint. However, we need first to show the minimizer’s existence, which shall demand lower semi-continuity and the coercivity of ${\mathcal{F}}_m^i,i=1,\dots ,4$.

Lemma 1.

The functionals ${\mathcal{F}}_m^i,i=1,\dots ,4$, defined in (17)–(20), respectively, are weakly lower and semi-continuous (WLS-C) in ${\mathrm{H}}_0^1(\Omega )$.

Proof. 

We suppose that $v_h\in {\mathrm{H}}_0^1(\Omega )$ converges weakly to v in ${\mathrm{H}}_0^1(\Omega )$ and take each term separately in the functionals ${\mathcal{F}}_m^i,i=1,\dots ,4$ defined in (17)–(20), respectively. Then we set the following:

$$\begin{array}{c}\hfill F^1\left(v\right)={\int }_{\Omega }{\displaystyle \frac{{\left|v-2v_{m-1}+v_{m-2}\right|}^2}{2k^2}}dx,\end{array}$$

(24)

$$\begin{array}{c}\hfill F^2\left(v\right)={\int }_{\Omega }\left({\displaystyle \frac{a}{2}}+{\displaystyle \frac{b}{4}}{∥\nabla v∥}_{{\mathrm{L}}^2(\Omega )}^2\right){\left|\nabla v\right|}^{2}dx,\end{array}$$

(25)

$$\begin{array}{c}\hfill F^3\left(v\right)={\int }_{\Omega }{F}_m\left(v\right)dx,\end{array}$$

(26)

$$\begin{array}{c}\hfill F^4\left(v\right)={\Psi }^1\left(v\right),\end{array}$$

(27)

$$\begin{array}{c}\hfill F^5\left(v\right)={\Psi }^2\left(v\right).\end{array}$$

(28)

We start with $F^1$, and we have that

$$\begin{array}{cc}\hfill 0\le \left|F^1\left(v_h\right)-F^1\left(v\right)\right|& ={\displaystyle \frac{1}{2k^2}}\left|{\int }_{\Omega }{\left(v_h-2v_{m-1}+v_{m-2}\right)}^2\mathrm{d}x-{\int }_{\Omega }{\left(v-2v_{m-1}+v_{m-2}\right)}^2\mathrm{d}x\right|\hfill \\ \hfill & ={\displaystyle \frac{1}{2k^2}}\left|{\int }_{\Omega }{\left(v_h\right)}^2-v^2+2\left(v_h-v\right)\left(-2v_{m-1}+v_{m-2}\right)\mathrm{d}x\right|\hfill \\ \hfill & \le {\displaystyle \frac{1}{2k^2}}{\int }_{\Omega }\left|v_h-v\right|\left|-4v_{m-1}+2v_{m-2}+v_h+v\right|\mathrm{d}x\hfill \\ \hfill & \le {\displaystyle \frac{1}{2k^2}}{∥v_h-v∥}_{{\mathrm{L}}^2(\Omega )}{∥v_h+v-4v_{m-1}+2v_{m-2}∥}_{{\mathrm{L}}^2(\Omega )}\to 0\mathrm{as}h\to \infty .\hfill \end{array}$$

(29)

while the last result is due to the Sobolev inequality, which leads to $F^1\left(v\right)=lim\; inf{F}^1\left(v_h\right)$. Then, we can say that $F^1$ is WLS-C in ${\mathrm{H}}_0^1(\Omega )$.

For $F^2$, by the use of the weakly lower semi-continuity of the norm in Banach space, it is clear that $F^2$ is WLS-C in ${\mathrm{H}}_0^1(\Omega )$.

Taking $F^3$, since $v_h\in {\mathrm{H}}_0^1(\Omega )$ and converges weakly to v in ${\mathrm{H}}_0^1(\Omega )$, and due to the continuity of $F_m$, we have that

$$F_m\left(v_h\right)\to F_m\left(v\right)\mathrm{as}h\to \infty \mathrm{a}.\mathrm{e}\mathrm{in}\Omega .$$

(30)

Furthermore, we have that

$$\left|F_m\left(v_h\right)\right|\le {\displaystyle \frac{C_f}{2}}{\left|v\right|}^2+{\delta }_m\left(x\right)\left|v\right|.$$

(31)

Then, by applying the DCT (dominated convergence theorem), we obtain

$$F^3\left(v_h\right)\to F^3\left(v\right)\mathrm{as}h\to \infty \mathrm{in}{\mathrm{H}}_0^1(\Omega ).$$

(32)

which means that $F^3$ is weakly lower and semi-continuous in ${\mathrm{H}}_0^1(\Omega )$.

For $F^4$, it is obvious that $F^4$ is WLS-C in ${\mathrm{H}}_0^1(\Omega )$. Concerning the lower semi-continuity of ${\Psi }^1$, since it takes only two values, the only problematic case can be if ${\Psi }^1\left(v_h\right)=\infty$ and ${lim\; inf}_{h\to \infty }{\Psi }^1\left(v_h\right)=0$. However, we can deal with this by contradiction. We suppose that ${\Psi }^1\left(v\right)=\infty$ and that $lim{inf}_{h\to \infty }{\Psi }^1\left(v_h\right)=0$. In this case, v is negative on positive measures set, say on $S\subset \Omega$. Hence, we obtain

$${\int }_{\Omega }{\left|v_h-v\right|}^2\mathrm{d}x={\int }_{\Omega \setminus S}{\left|v_h-v\right|}^2\mathrm{d}x+{\int }_S{\left|v_h-v\right|}^2\mathrm{d}x.$$

(33)

Based on the Sobolev inequality, the previous has to converge to zero as h does to infinity. Nonetheless, v is negative on S and for any h there exists $H\ge h$ such that ${\Psi }^2\left(v_h\right)=0$. Then, the last term will not vanish in the limit, leading to a contradiction with convergence.

Similar to $F^4$, for $F^5$, the only questionable case is if ${\Psi }^2\left(v\right)=\infty$ and ${lim\; inf}_{h\to \infty }{\Psi }^2\left(v_h\right)=0$. Again, for any h, there always exists $H\ge h$ such that ${\int }_{\Omega }{v}_h\mathrm{d}x=V$. Then, we get the chain of inequalities as follows:

$$0<\left|{\int }_{\Omega }\left(v_h-v\right)\mathrm{d}x\right|\le {\int }_{\Omega }\left|v_h-v\right|\mathrm{d}x\le {|\Omega |}^{1/2}{\left({\int }_{\Omega }{\left|v_h-v\right|}^2\mathrm{d}x\right)}^{1/2}.$$

(34)

The Sobolev inequality again implies that the right-hand side approaches zero as $H\to \infty$, which contradicts the strict inequality.

Since $F^i;i=1,\dots ,5$ are WLS-C functions, then ${\mathcal{F}}_m^1$ and ${\mathcal{F}}_m^2$ are WLS-C in ${\mathrm{H}}_0^1(\Omega )$. □

Lemma 2.

The functionals ${\mathcal{F}}_m^i,i=1,\dots ,4$ defined in (17)–(20), respectively, are coercive in ${\mathrm{H}}_0^1(\Omega )$.

Proof. 

Choosing a minimizing sequence ${\left\{v_h\right\}}_{h=1}^{\infty }$ in ${\mathrm{H}}_0^1(\Omega )$ converges weakly to v in ${\mathrm{H}}_0^1(\Omega )$, moreover ${\displaystyle {\mathcal{F}}_m^1\left(v\right)=\underset{h\to \infty }{lim\; inf}{\mathcal{F}}_m^1\left(v_h\right),\mathrm{and}{\mathcal{F}}_m^1\left(v_1\right)\ne \infty .}$ Then, we have that

$$\begin{array}{cc}\hfill {\mathcal{F}}_m^1\left(v_h\right)& ={\int }_{\Omega }\left[{\displaystyle \frac{{\left|v-2v_{m-1}+v_{m-2}\right|}^2}{2k^2}}+\left({\displaystyle \frac{a}{2}}+{\displaystyle \frac{b}{4}}{∥\nabla v∥}_{{\mathrm{L}}^2(\Omega )}^2\right){\left|\nabla v\right|}^2+F_m\left(v\right)\right]dx\hfill \\ & \ge {\int }_{\Omega }\left[\left({\displaystyle \frac{a}{2}}+{\displaystyle \frac{b}{4}}{∥\nabla v_h∥}_{{\mathrm{L}}^2(\Omega )}^2\right){\left|\nabla v_h\right|}^2-F_m\left(v_h\right)\right]dx\hfill \\ & \ge {\int }_{\Omega }\left[\left({\displaystyle \frac{a}{2}}+{\displaystyle \frac{b}{4}}{∥\nabla v_h∥}_{{\mathrm{L}}^2(\Omega )}^2\right){\left|\nabla v_h\right|}^2-{\displaystyle \frac{C_f}{2}}{\left|v_h\right|}^2-{\delta }_m{v}_h\right]dx\hfill \\ & \ge \left({\displaystyle \frac{a}{2}}+{\displaystyle \frac{b}{4}}{∥\nabla v_h∥}_{{\mathrm{L}}^2(\Omega )}^2\right){∥\nabla v_h∥}_{{\mathrm{L}}^2(\Omega )}^2-{\displaystyle \frac{C_f}{2}}{∥v_h∥}_{{\mathrm{L}}^2(\Omega )}^2-{\displaystyle \frac{1}{2}}{∥{\delta }_m∥}_{{\mathrm{L}}^2(\Omega )}^2-{\displaystyle \frac{1}{2}}{∥v_h∥}_{{\mathrm{L}}^2(\Omega )}^2\hfill \\ & \ge \left({\displaystyle \frac{a}{2}}+{\displaystyle \frac{b}{4}}{∥\nabla v_h∥}_{{\mathrm{L}}^2(\Omega )}^2\right){∥\nabla v_h∥}_{{\mathrm{L}}^2(\Omega )}^2-{\displaystyle \frac{C_p^2{C}_f}{2}}{∥\nabla v_h∥}_{\mathrm{L}2(\Omega )}^2-{\displaystyle \frac{1}{2}}{∥{\delta }_m∥}_{{\mathrm{L}}^2(\Omega )}^2-{\displaystyle \frac{C_p^2}{2}}{∥\nabla v_h∥}_{{\mathrm{L}}^2(\Omega )}^2\hfill \\ & ={\displaystyle \frac{b}{4}}{∥\nabla v_h∥}_{{\mathrm{L}}^2(\Omega )}^4+\left({\displaystyle \frac{a}{2}}-{\displaystyle \frac{C_p^2{C}_f}{2}}-{\displaystyle \frac{C_p^2}{2}}\right){∥\nabla v_h∥}_{{\mathrm{L}}^2(\Omega )}^2-{\displaystyle \frac{1}{2}}{∥{\delta }_m∥}_{{\mathrm{L}}^2(\Omega )}^2\hfill \end{array}$$

(35)

where $C_p$ is the Poincaré coefficient and ${∥{\delta }_m∥}_{{\mathrm{L}}^2(\Omega )}^2<\infty .$ Then, we have that

$${\mathcal{F}}_m^1\left(v_h\right)\to \infty \mathrm{as}{∥\nabla v_h∥}_{{\mathrm{L}}^2(\Omega )}\to \infty \mathrm{in}{\mathrm{L}}^2(\Omega ),$$

(36)

and since

$${∥\nabla v_h∥}_{{\mathrm{L}}^2(\Omega )}\ge C{∥v_h∥}_{{\mathrm{H}}_0^1(\Omega )}$$

(37)

then

$${\mathcal{F}}_m^1\left(v_h\right)\to \infty \mathrm{as}{∥v_h∥}_{{\mathrm{H}}_0^1(\Omega )}\to \infty \mathrm{in}{\mathrm{H}}_0^1(\Omega ).$$

(38)

Following the same steps as (35) and since ${\Psi }^1$ and ${\Psi }^2$ are positive, we can obtain that

$${\mathcal{F}}_m^i\left(v_h\right)\to \infty \mathrm{as}{∥v_h∥}_{{\mathrm{H}}_0^1(\Omega )}\to \infty \mathrm{in}{\mathrm{H}}_0^1(\Omega ),\mathrm{for}i=2,\dots ,4.$$

(39)

then the functionals ${\mathcal{F}}_m^i,i=1,\dots ,4$ defined in (17)–(20), respectively, are coercive in ${\mathrm{H}}_0^1(\Omega )$. □

Lemma 3.

Let v be in ${\mathrm{H}}_0^1(\Omega )$, hence v is a minimizer of the functionals ${\mathcal{F}}_m^i,i=1,\dots ,4$ defined in (17)–(20), respectively, in ${\mathrm{H}}_0^1(\Omega )$.

Proof. 

Since the functionals ${\mathcal{F}}_m^i,i=1,\dots ,4$ defined in (17)–(20), respectively, are coercive and WLS-C in ${\mathrm{H}}_0^1(\Omega )$, then ${\mathcal{F}}_n$ has a minimizer in ${\mathrm{H}}_0^1(\Omega )$.

Furthermore, in the same manner as (35), we can obtain that

$${\mathcal{F}}_m^i\left(v_1\right)\ge {∥v_h∥}_{{\mathrm{H}}_0^1(\Omega )},i=1,\dots ,4.$$

(40)

This boundedness enables us to extract a subsequence noted again as $v_h$ that is weakly convergent, such that

$$v_h\rightharpoonup v\mathrm{in}{\mathrm{H}}_0^1(\Omega ).$$

(41)

Then v is a minimizer of the functionals ${\mathcal{F}}_m^i,i=1,\dots ,4$ defined in (17)–(20), respectively, in ${\mathrm{H}}_0^1(\Omega )$. □

Next, we define piecewise linear time interpolant solutions $v^k$ for $(x,t)\in \Omega \times \left((h-1)k,kh\right],$

where $h=0,1,\dots ,M$ by

$$v^k(x,t)={\displaystyle \frac{(t-(h-1\left)k\right)}{k}}{v}_h\left(x\right)+{\displaystyle \frac{kh-t}{k}}{v}_{h-1}\left(x\right),$$

(42)

and a piecewise constant step function ${\overline{v}}^k$ by

$${\overline{v}}^k(x,t)=v_h\left(x\right).$$

(43)

In the same way, we define $(x,t)\in \Omega \times \left((h-1)k,kh\right]$ where $h=0,1,\dots ,M$ by

$${\overline{f}}^k(x,t,v)=f_h(x,v).$$

(44)

In the context of the defined piecewise linear and constant functions, the following lemma characterizes the approximate weak solution within the specified function spaces.

Lemma 4.

For all $\varphi (x,t)\in C_0^{\infty }(\Omega \times [0,T))$, we observe that

$$\begin{array}{cc}\hfill & {\int }_0^T{\int }_{\Omega }{\displaystyle \frac{v_t^k(x,t)-v_t^k(x,t-k)}{k}}\varphi (x,t)\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & +{\int }_0^T{\int }_{\Omega }\left(a+b{∥\nabla {\overline{v}}^k(x,t)∥}_{{\mathrm{L}}^2(\Omega )}^2\right){\left|\nabla {\overline{v}}^k(x,t)\right|}^{p-2}\nabla {\overline{v}}^k(x,t)\nabla \varphi (x,t)\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & +{\int }_0^T{\int }_{\Omega }{\overline{f}}^k\left(x,t,{\overline{v}}^k\right)\varphi (x,t)\mathrm{d}x\mathrm{d}t=0.\hfill \end{array}$$

(45)

That is an approximate weak solution to the problem defined in (1), and if $\varphi (x,t)$ is with zero volume (i.e., ${\int }_{\Omega }\varphi (x,t)=0$), the approximation (45) is an approximate weak solution to the problem defined in (3), while the approximation (45) is an approximate weak solution to the problem defined in (2) if $\varphi (x,t)\in C_0^{\infty }(\Omega \times [0,T)\cap \left\{{\overline{v}}^k>0\right\})$. Lastly, the approximate weak solution to the problem defined in (1) is the approximation (45) for $\varphi (x,t)\in C_0^{\infty }(\Omega \times [0,T)\cap \left\{{\overline{v}}^k>0\right\})$ with zero volume.

Proof. 

Let $v_m$ be a minimizer of ${\mathcal{F}}_m^1$ then, for $ϵ>0$, and all $\varphi (x,t)\in C_0^{\infty }((\Omega \times [0,T))$, we have

$${\mathcal{F}}_m^1(v_m+ϵ\varphi )\ge {\mathcal{F}}_m^1\left(v_m\right).$$

(46)

Hence, we can write

$$\begin{array}{cc}\hfill 0& =\underset{ϵ\to 0^{+}}{lim}{\displaystyle \frac{{\Psi }^1\left(v_m+ϵ\varphi \right)-{\Psi }^1\left(v_m\right)}{ϵ}}\hfill \\ \hfill & \ge -{\int }_{\Omega }{\displaystyle \frac{v_m-2v_{m-1}+v_{m-2}}{k^2}}\varphi \mathrm{d}x-{\int }_{\Omega }\left(a+b{∥\nabla v_m(x,t)∥}_{{\mathrm{L}}^2(\Omega )}^2\right)\nabla v_m\nabla \varphi \mathrm{d}x-{\int }_{\Omega }{f}_m\left(v_m\right)\varphi \mathrm{d}x.\hfill \end{array}$$

(47)

Taking $-\varphi$, and by the use of the definition of $v^k$ in (42) and ${\overline{v}}^k$ in (43), this yields the opposite inequality

$$0={\int }_{\Omega }{\displaystyle \frac{v_t^k\left(t\right)-v_t^k(t-k)}{k}}\varphi \mathrm{d}x+{\int }_{\Omega }\left(a+b{∥\nabla {\overline{v}}^k(x,t)∥}_{{\mathrm{L}}^2(\Omega )}^2\right)\nabla {\overline{v}}^k\nabla \varphi \mathrm{d}x+{\int }_{\Omega }{\overline{f}}^k\left({\overline{v}}^k\right)\varphi \mathrm{d}x.$$

(48)

Almost everywhere, $t\in (0,T)$ for all $\varphi \in C_0^{\infty }\left(\Omega \right)$. By integrating over $(0,T)$ and extending the test functions to time-dependent domains, we obtain

$$\begin{array}{cc}\hfill {\int }_0^T& {\int }_{\Omega }\left({\displaystyle \frac{v_t^k(x,t)-v_t^k(x,t-k)}{k}}\varphi (x,t)+\left(a+b{∥\nabla {\overline{v}}^k(x,t)∥}_{{\mathrm{L}}^2(\Omega )}^2\right)\nabla {\overline{v}}^k(x,t)\nabla \varphi (x,t)\right)\mathrm{d}x\mathrm{d}t\hfill \\ \hfill +& {\int }_0^T{\int }_{\Omega }{\overline{f}}^k\left(x,t,{\overline{v}}^k\right)\varphi (x,t)\mathrm{d}x\mathrm{d}t=0,\hfill \end{array}$$

(49)

for all $\varphi (x,t)\in C_0^{\infty }(\Omega \times [0,T))$. Note that we obtain the same approximate weak solution according to Definitions 2, 3, and 4 using ${\mathcal{F}}_m^2$, ${\mathcal{F}}_m^3$, and ${\mathcal{F}}_m^4$, respectively; the only difference is that $\varphi (x,t)\in C_0^{\infty }(\Omega \times [0,T)\cap \left\{{\overline{v}}^k>0\right\})$, $\varphi (x,t)\in C_0^{\infty }(\Omega \times [0,T))$ with zero volume, and $\varphi (x,t)\in C_0^{\infty }(\Omega \times [0,T)\cap \left\{{\overline{v}}^k>0\right\})$ with zero volume, respectively. □

Next, we will refine the time step as k approaches zero and look for the convergence. Therefore, by passing the limit, we can prove the weak solution’s existence. For this, an energy estimation is required.

3. Energy Estimate

In this section, we present an energy estimate for minimizers that offers crucial insights into their convergence. The following lemma shows why this estimate is important for understanding their convergence.

Lemma 5.

The functions $v^k$ and ${\overline{v}}^k$ obey the estimate

$${∥v_t^k\left(t\right)∥}_{{\mathrm{L}}^2(\Omega )}^2+{∥{\overline{v}}^k\left(t\right)∥}_{{\mathrm{H}}_0^1(\Omega )}^2\le C,$$

(50)

where C is an independent constant of k, such that

$$C=\left({\displaystyle \frac{1}{2}}{∥u_0∥}_{{\mathbb{L}}^2(\Omega )}^2+\left({\displaystyle \frac{a}{4}}+{\displaystyle \frac{b}{8}}{∥\nabla v_0∥}_{{\mathbb{L}}^2(\Omega )}^2\right){∥\nabla v_0∥}_{{\mathbb{L}}^2(\Omega )}^2+{∥\gamma ∥}_{{\mathbb{L}}^2(\Omega )}^2\right)e^{max\left\{c_p^2,1\right\}T}.$$

(51)

Proof. 

Let $v_h$ be a minimizer of ${\mathcal{F}}_h^i,i=1,\dots ,4$ and for an arbitrary $\theta \in (0,1)$, we have that

$${\mathcal{F}}_h^i\left(v_h\right)\le {\mathcal{F}}_h^i\left((1-\theta )v_h+\theta \left(v_{h-1}\right)\right),i=1,\dots ,4.$$

(52)

Thus,

$$0\le \underset{\theta \to 0^{+}}{lim}{\displaystyle \frac{{\mathcal{F}}_h^i\left((1-\theta )v_h+\theta \left(v_{h-1}\right)\right)-{\mathcal{F}}_h^i\left(v_h\right)}{\theta }},i=1,\dots ,4.$$

(53)

We investigate each term in the above alone. Then we set the following:

$$\begin{array}{c}\hfill F^1\left(\theta \right)=\underset{\theta \to 0^{+}}{lim}{\int }_{\Omega }{\displaystyle \frac{{\left|(1-\theta )v_h+\theta v_{h-1}-2v_{h-1}+v_{h-2}\right|}^2-{\left|v_h-2v_{h-1}+v_{h-2}\right|}^2}{2h^2\theta }}\mathrm{d}x,\end{array}$$

(54)

$$\begin{array}{c}\hfill F^2\left(\theta \right)=\underset{\theta \to 0^{+}}{lim}{\int }_{\Omega }{\displaystyle \frac{\left({\displaystyle \frac{a}{2}}+{\displaystyle \frac{b}{4}}{∥\nabla \left((1-\theta )v_h+\theta v_{h-1}\right)∥}_{{\mathrm{L}}^2(\Omega )}^2\right){\left|\nabla \left((1-\theta )v_h+\theta v_{h-1}\right)\right|}^2}{\theta }}\mathrm{d}x\end{array}$$

(55)

$$\begin{array}{c}\hfill -{\int }_{\Omega }{\displaystyle \frac{\left({\displaystyle \frac{a}{2}}+{\displaystyle \frac{b}{4}}{∥\nabla v_h∥}_{{\mathrm{L}}^2(\Omega )}^2\right){\left|\nabla v_h\right|}^2}{\theta }}\mathrm{d}x,\end{array}$$

(56)

$$\begin{array}{c}\hfill F^3\left(\theta \right)=\underset{\theta \to 0^{+}}{lim}{\int }_{\Omega }{\displaystyle \frac{F_h\left((1-\theta )v_h+\theta v_{h-1}\right)-F_h\left(v_h\right)}{\theta }}\mathrm{d}x,\end{array}$$

(57)

$$\begin{array}{c}\hfill F^4\left(\theta \right)=\underset{\theta \to 0^{+}}{lim}{\displaystyle \frac{{\Psi }_i\left((1-\theta )v_h+\theta v_{h-1}\right)-{\Psi }_i\left(v_h\right)}{2\theta }};i=1,2.\end{array}$$

(58)

We start with $F^1\left(\theta \right)$, with direct calculation, and using the following inequality from [10]

$${\left|a\right|}^{p-2}a(b-a)\le {\displaystyle \frac{{\left|b\right|}^p-{\left|a\right|}^p}{p}},\mathrm{where}a,b\in \mathbb{R},$$

(59)

and for $p=2$, we obtain

$$a(b-a)\le {\displaystyle \frac{b^2-a^2}{2}}.$$

(60)

We have that

$$\begin{array}{cc}\hfill F^1\left(\theta \right)=& \underset{\theta \to 0^{+}}{lim}{\int }_{\Omega }{\displaystyle \frac{{\left|(1-\theta )v_h+\theta v_{h-1}-2v_{h-1}+v_{h-2}\right|}^2-{\left|v_h-2v_{h-1}+v_{h-2}\right|}^2}{2k^2\theta }}\mathrm{d}x\hfill \\ \hfill & ={\int }_{\Omega }{\displaystyle \frac{\left(v_{h-1}-v_h\right)\left(v_h-2v_{h-1}+v_{h-2}\right)}{k^2}}dx\hfill \\ \hfill & ={\int }_{\Omega }{\displaystyle \frac{\left(v_{h-1}-v_h\right)\left((v_h-v_{h-1})-(v_{h-1}-v_{h-2})\right)}{k^2}}dx\hfill \\ \hfill & \le {\int }_{\Omega }{\displaystyle \frac{{\left|v_{h-2}-v_{h-1}\right|}^2-{\left|v_{h-1}-v_h\right|}^2}{2k^2}}dx\hfill \\ \hfill & ={\displaystyle \frac{1}{2k^2}}\left({∥v_{h-1}-v_{h-2}∥}_{{\mathrm{L}}^2(\Omega )}^2-{∥v_h-v_{h-1}∥}_{{\mathrm{L}}^2(\Omega )}^2\right).\hfill \end{array}$$

(61)

Next, using (59), we obtain the next estimation for $F^2\left(\theta \right)$

$$\begin{array}{cc}\hfill F^2\left(\theta \right)=& \underset{\theta \to 0^{+}}{lim}{\int }_{\Omega }{\displaystyle \frac{\left(\frac{a}{2}+\frac{b}{4}{∥\nabla \left((1-\theta )v_h+\theta v_{h-1}\right)∥}_{{\mathbb{L}}^2(\Omega )}^2\right){\left|\nabla \left((1-\theta )v_h+\theta v_{h-1}\right)\right|}^2}{2\theta }}\mathrm{d}x\hfill \\ & +\underset{\theta \to 0^{+}}{lim}{\int }_{\Omega }{\displaystyle \frac{\left(\frac{a}{2}+\frac{b}{4}{∥\nabla v_h∥}_{{\mathbb{L}}^2(\Omega )}^2\right){\left|\nabla v_h\right|}^2}{2\theta }}\mathrm{d}x\hfill \\ \hfill & ={\displaystyle \frac{b}{4}}{∥\nabla v_h∥}_{{\mathbb{L}}^2(\Omega )}^2{\int }_{\Omega }\nabla v_h\left(\nabla v_{h-1}-\nabla v_h\right)\mathrm{d}x\hfill \\ \hfill & +\left({\displaystyle \frac{a}{2}}+{\displaystyle \frac{b}{4}}{∥\nabla v_h∥}_{{\mathbb{L}}^2(\Omega )}^2\right){\int }_{\Omega }\nabla v_h\left(\nabla v_{h-1}-\nabla v_h\right)\mathrm{d}x\hfill \\ \hfill & \le {\displaystyle \frac{b}{4}}{∥\nabla v_h∥}_{{\mathbb{L}}^2(\Omega )}^2{\int }_{\Omega }{\displaystyle \frac{{\left|\nabla v_{h-1}\right|}^2-{\left|\nabla v_h\right|}^2}{2}}\mathrm{d}x\hfill \\ \hfill & +\left({\displaystyle \frac{a}{2}}+{\displaystyle \frac{b}{4}}{∥\nabla v_h∥}_{{\mathbb{L}}^2(\Omega )}^2\right){\int }_{\Omega }{\displaystyle \frac{{\left|\nabla v_{h-1}\right|}^2-{\left|\nabla v_h\right|}^2}{2}}\mathrm{d}x\hfill \\ \hfill & \le {\displaystyle \frac{1}{2}}{\displaystyle \frac{b}{4}}{∥\nabla v_h∥}_{{\mathbb{L}}^2(\Omega )}^2\left({∥\nabla v_{h-1}∥}^2-{∥\nabla v_h∥}^2\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}\left({\displaystyle \frac{a}{2}}+{\displaystyle \frac{b}{4}}{∥\nabla v_h∥}_{{\mathbb{L}}^2(\Omega )}^2\right)\left({∥\nabla v_{h-1}∥}^2-{∥\nabla v_h∥}^2\right)\hfill \\ \hfill & =\left({\displaystyle \frac{a}{4}}+{\displaystyle \frac{b}{4}}{∥\nabla v_h∥}_{{\mathbb{L}}^2(\Omega )}^2\right)\left({∥\nabla v_{h-1}∥}_{{\mathbb{L}}^2(\Omega )}^2-{∥\nabla v_h∥}_{{\mathbb{L}}^2(\Omega )}^2\right).\hfill \end{array}$$

(62)

We obtain another estimate for $F^3\left(\theta \right)$ by using (7), as in the following

$$\begin{array}{cc}\hfill F^3\left(\theta \right)=& \underset{\theta \to 0^{+}}{lim}{\int }_{\Omega }{\displaystyle \frac{F_h\left((1-\theta )v_h+\theta v_{h-1}\right)-F_h\left(v_h\right)}{\theta }}\mathrm{d}x\hfill \\ \hfill & =\underset{\theta \to 0^{+}}{lim}{\int }_{\Omega }{\displaystyle \frac{F_h\left(v_h+\theta (v_{h-1}-v_h)\right)-F_h\left(v_h\right)}{\theta }}\mathrm{d}x\hfill \\ \hfill & ={\int }_{\Omega }{f}_h\left(v_h\right)\left(v_{h-1}-v_h\right)\mathrm{d}x\hfill \\ \hfill & \le {\int }_{\Omega }\left(C_f\left|v_h\right|+{\gamma }_h\right)\left|v_h-v_{h-1}\right|\mathrm{d}x\hfill \\ \hfill & \le {\int }_{\Omega }{\displaystyle \frac{k}{2}}{\left(C_f\left|v_h\right|+{\gamma }_h\right)}^2+{\displaystyle \frac{1}{2k}}{\left|v_h-v_{h-1}\right|}^2\mathrm{d}x\hfill \\ \hfill & \le C_f^{2}k{\int }_{\Omega }{v}_h^2\mathrm{d}x+k{\int }_{\Omega }{\gamma }_h^2\mathrm{d}x+{\displaystyle \frac{k}{2}}{\int }_{\Omega }{\left({\displaystyle \frac{v_h-v_{h-1}}{k}}\right)}^2\mathrm{d}x\hfill \\ \hfill & =C_f^{2}k{∥v_h∥}_{{\mathbb{L}}^2(\Omega )}^2+k{∥{\gamma }_h∥}_{{\mathbb{L}}^2(\Omega )}^2+k{∥{\displaystyle \frac{v_h-v_{h-1}}{2k}}∥}_{{\mathbb{L}}^2(\Omega )}^2.\hfill \end{array}$$

(63)

For $F^4\left(\theta \right)$, we have that

$${\displaystyle {\int }_{\Omega }\left((1-\theta )v_h+\theta \left(v_{h-1}\right)\right)dx=V,}$$

(64)

and

$$(1-\theta )v_h+\theta \left(v_{h-1}\right)\ge 0\mathrm{for}\mathrm{a}.\mathrm{e}.x\in \Omega .$$

(65)

Then, it is clear that

$$\underset{\theta \to 0^{+}}{lim}{\displaystyle \frac{{\Psi }_i\left((1-\theta )v_h+\theta v_{h-1}\right)-{\Psi }_i\left(v_h\right)}{2\theta }}=0,\mathrm{for}i=1,2.$$

(66)

Summing the result obtained from $F^1,F^2,F^3$ and $F^4$, we obtain the following energy estimate for the minimizers of ${\mathcal{F}}_m^i,i=1,\dots ,4$, and as following:

$$\begin{array}{cc}\hfill 0& \le {\displaystyle \frac{1}{2k^2}}\left({∥v_{h-1}-v_{h-2}∥}_{{\mathbb{L}}^2(\Omega )}^2-{∥v_h-v_{h-1}∥}_{{\mathbb{L}}^2(\Omega )}^2\right)\hfill \\ \hfill & +\left({\displaystyle \frac{a}{4}}+{\displaystyle \frac{b}{4}}{∥\nabla v_h∥}_{{\mathbb{L}}^2(\Omega )}^2\right)\left({∥\nabla v_{h-1}∥}_{{\mathbb{L}}^2(\Omega )}^2-{∥\nabla v_h∥}_{{\mathbb{L}}^2(\Omega )}^2\right)\hfill \\ \hfill & +C_f^{2}k{∥v_h∥}_{{\mathbb{L}}^2(\Omega )}^2+k{∥{\gamma }_h∥}_{{\mathbb{L}}^2(\Omega )}^2+{\displaystyle \frac{k}{2}}{∥{\displaystyle \frac{v_h-v_{h-1}}{k}}∥}_{{\mathbb{L}}^2(\Omega )}^2.\hfill \end{array}$$

(67)

This leads to

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{2k^2}}\left({∥v_h-v_{h-1}∥}_{{\mathbb{L}}^2\Omega }^2-{∥v_{h-1}-v_{h-2}∥}_{{\mathbb{L}}^2(\Omega )}^2\right)\hfill \\ \hfill & +\left({\displaystyle \frac{a}{4}}+{\displaystyle \frac{b}{4}}{∥\nabla v_h∥}_{{\mathbb{L}}^2(\Omega )}^2\right)\left({∥\nabla v_h∥}_{{\mathbb{L}}^2(\Omega )}^2-{∥\nabla v_{h-1}∥}_{{\mathbb{L}}^2(\Omega )}^2\right)\hfill \\ \hfill & \le k{c}_p^2{C}_f^2{∥\nabla v_h∥}_{{\mathbb{L}}^2(\Omega )}^2+k{∥{\gamma }_h∥}_{{\mathbb{L}}^2(\Omega )}^2+{\displaystyle \frac{k}{2}}{∥{\displaystyle \frac{v_h-v_{h-1}}{k}}∥}_{{\mathbb{L}}^2(\Omega )}^2.\hfill \end{array}$$

(68)

By utilizing (60), we simplify the Kirchhoff term to obtain

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{2k^2}}\left({∥v_h-v_{h-1}∥}_{{\mathbb{L}}^2\Omega }^2-{∥v_{h-1}-v_{h-2}∥}_{{\mathbb{L}}^2(\Omega )}^2\right)+{\displaystyle \frac{a}{4}}\left({∥\nabla v_h∥}_{{\mathbb{L}}^2(\Omega )}^2-{∥\nabla v_{h-1}∥}_{{\mathbb{L}}^2(\Omega )}^2\right)\hfill \\ \hfill & +{\displaystyle \frac{b}{8}}\left({∥\nabla v_h∥}_{{\mathbb{L}}^2(\Omega )}^4-{∥\nabla v_{h-1}∥}_{{\mathbb{L}}^2(\Omega )}^4\right)\le k{c}_p^2{C}_f^2{∥\nabla v_h∥}_{{\mathbb{L}}^2(\Omega )}^2+k{∥{\gamma }_h∥}_{{\mathbb{L}}^2(\Omega )}^2+{\displaystyle \frac{k}{2}}{∥{\displaystyle \frac{v_h-v_{h-1}}{k}}∥}_{{\mathbb{L}}^2(\Omega )}^2.\hfill \end{array}$$

(69)

Summing with respect to h from 1 to m results in a telescoping series with accumulation, leading to

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{2k^2}}\left({∥v_m-v_{m-1}∥}_{{\mathbb{L}}^2(\Omega )}^2-{∥v_0-v_1∥}_{{\mathbb{L}}^2(\Omega )}^2\right)+{\displaystyle \frac{a}{4}}\left({∥\nabla v_m∥}_{{\mathbb{L}}^2(\Omega )}^2-{∥\nabla v_0∥}_{{\mathbb{L}}^2(\Omega )}^2\right)\hfill \\ \hfill & +{\displaystyle \frac{b}{8}}\left({∥\nabla v_m∥}_{{\mathbb{L}}^2(\Omega )}^4-{∥\nabla v_0∥}_{{\mathbb{L}}^2(\Omega )}^4\right)\le \sum _{h=1}^{m}k{c}_p^2{C}_f^2{∥\nabla v_h∥}_{{\mathbb{L}}^2(\Omega )}^2+k{∥{\gamma }_h∥}_{{\mathbb{L}}^2(\Omega )}^2+{\displaystyle \frac{k}{2}}{∥{\displaystyle \frac{v_h-v_{h-1}}{k}}∥}_{{\mathbb{L}}^2(\Omega )}^2.\hfill \end{array}$$

(70)

We have that ${\displaystyle \frac{v_0-v_1}{k}}=u_0$, so the above inequality can be rewritten as follows

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{2k^2}}{∥v_m-v_{m-1}∥}_{{\mathbb{L}}^2(\Omega )}^2+\left({\displaystyle \frac{a}{4}}+{\displaystyle \frac{b}{8}}{∥\nabla v_m∥}_{{\mathbb{L}}^2(\Omega )}^2\right){∥\nabla v_m∥}_{{\mathbb{L}}^2(\Omega )}^2\hfill \\ \hfill & \le {\displaystyle \frac{1}{2}}{∥u_0∥}_{{\mathbb{L}}^2(\Omega )}^2+\left({\displaystyle \frac{a}{4}}+{\displaystyle \frac{b}{8}}{∥\nabla v_0∥}_{{\mathbb{L}}^2(\Omega )}^2\right){∥\nabla v_0∥}_{{\mathbb{L}}^2(\Omega )}^2+{∥\gamma ∥}_{{\mathbb{L}}^2(\Omega )}^2\hfill \\ & +\sum _{h=1}^{m}k\left(c_p^2{C}_f^2{∥\nabla v_h∥}_{{\mathbb{L}}^2(\Omega )}^2+{\displaystyle \frac{1}{2}}{∥{\displaystyle \frac{v_h-v_{h-1}}{k}}∥}_{{\mathbb{L}}^2(\Omega )}^2\right).\hfill \end{array}$$

(71)

As the components in the left term of the inequality are positive, we have

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{2k^2}}{∥v_m-v_{m-1}∥}_{{\mathbb{L}}^2(\Omega )}^2+{\displaystyle \frac{a}{4}}{∥\nabla v_m∥}_{{\mathbb{L}}^2(\Omega )}^2\hfill \\ \hfill & \le {\displaystyle \frac{1}{2}}{∥u_0∥}_{{\mathbb{L}}^2(\Omega )}^2+\left({\displaystyle \frac{a}{4}}+{\displaystyle \frac{b}{8}}{∥\nabla v_0∥}_{{\mathbb{L}}^2(\Omega )}^2\right){∥\nabla v_0∥}_{{\mathbb{L}}^2(\Omega )}^2+{∥\gamma ∥}_{{\mathbb{L}}^2(\Omega )}^2\hfill \\ & +\sum _{h=1}^{m}k\left(c_p^2{C}_f^2{∥\nabla v_h∥}_{{\mathbb{L}}^2(\Omega )}^2+{\displaystyle \frac{1}{2}}{∥{\displaystyle \frac{v_h-v_{h-1}}{k}}∥}_{{\mathbb{L}}^2(\Omega )}^2\right).\hfill \end{array}$$

(72)

From (6), we have that

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{2k^2}}{∥v_m-v_{m-1}∥}_{{\mathbb{L}}^2(\Omega )}^2+{\displaystyle \frac{a}{4}}{∥\nabla v_m∥}_{{\mathbb{L}}^2(\Omega )}^2\hfill \\ \hfill & \le {\displaystyle \frac{1}{2}}{∥u_0∥}_{{\mathbb{L}}^2(\Omega )}^2+\left({\displaystyle \frac{a}{4}}+{\displaystyle \frac{b}{8}}{∥\nabla v_0∥}_{{\mathbb{L}}^2(\Omega )}^2\right){∥\nabla v_0∥}_{{\mathbb{L}}^2(\Omega )}^2+{∥\gamma ∥}_{{\mathbb{L}}^2(\Omega )}^2\hfill \\ & +\sum _{h=1}^{m}k\left(c_p^2{\displaystyle \frac{a}{4}}{∥\nabla v_h∥}_{{\mathbb{L}}^2(\Omega )}^2+{\displaystyle \frac{1}{2}}{∥{\displaystyle \frac{v_h-v_{h-1}}{k}}∥}_{{\mathbb{L}}^2(\Omega )}^2\right),\hfill \end{array}$$

(73)

which leads to

$$\begin{array}{cc}\hfill & {\displaystyle \frac{1}{2k^2}}{∥v_m-v_{m-1}∥}_{{\mathbb{L}}^2(\Omega )}^2+{\displaystyle \frac{a}{4}}{∥\nabla v_m∥}_{{\mathbb{L}}^2(\Omega )}^2\hfill \\ \hfill & \le {\displaystyle \frac{1}{2}}{∥u_0∥}_{{\mathbb{L}}^2(\Omega )}^2+\left({\displaystyle \frac{a}{4}}+{\displaystyle \frac{b}{8}}{∥\nabla v_0∥}_{{\mathbb{L}}^2(\Omega )}^2\right){∥\nabla v_0∥}_{{\mathbb{L}}^2(\Omega )}^2+{∥\gamma ∥}_{{\mathbb{L}}^2(\Omega )}^2\hfill \\ & +max\left\{c_p^2,1\right\}\sum _{h=1}^{m}k\left({\displaystyle \frac{a}{4}}{∥\nabla v_h∥}_{{\mathbb{L}}^2(\Omega )}^2+{\displaystyle \frac{1}{2}}{∥{\displaystyle \frac{v_h-v_{h-1}}{k}}∥}_{{\mathbb{L}}^2(\Omega )}^2\right).\hfill \end{array}$$

(74)

Then, employing the discrete Grönwall lemma, we derive

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{2k^2}}{∥v_m-v_{m-1}∥}_{{\mathbb{L}}^2(\Omega )}^2+{\displaystyle \frac{a}{4}}{∥\nabla v_m∥}_{{\mathbb{L}}^2(\Omega )}^2\le C,\end{array}$$

(75)

where

$$C=\left({\displaystyle \frac{1}{2}}{∥u_0∥}_{{\mathbb{L}}^2(\Omega )}^2+\left({\displaystyle \frac{a}{4}}+{\displaystyle \frac{b}{8}}{∥\nabla v_0∥}_{{\mathbb{L}}^2(\Omega )}^2\right){∥\nabla v_0∥}_{{\mathbb{L}}^2(\Omega )}^2+{∥\gamma ∥}_{{\mathbb{L}}^2(\Omega )}^2\right)e^{max\left\{c_p^2,1\right\}T}$$

(76)

This is equivalent to

$$\begin{array}{c}\hfill {\displaystyle \frac{1}{2}}{∥v_t^k∥}_{{\mathbb{L}}^2(\Omega )}^2+{\displaystyle \frac{a}{4}}{∥\nabla {\overline{v}}^k∥}_{{\mathbb{L}}^2(\Omega )}^2\le C,\end{array}$$

(77)

which implies that

$${∥v_t^k(x,t)∥}_{{\mathbb{L}}^2(\Omega )}^2+{∥{\overline{v}}^k(x,t)∥}_{{\mathbb{H}}_0^1(\Omega )}^2\le C.$$

(78)

□

The above estimate leads immediately to the next lemma.

Lemma 6.

A function v exists, such that $v\in {\mathrm{L}}^{\infty }(0,T;{\mathrm{H}}_0^1(\Omega ))\cap {\mathrm{W}}^{1,\infty }(0,T;{\mathrm{L}}^2(\Omega ))$, and a sub-sequence denoted by $v^k$ again, such that

$$\begin{array}{cc}\hfill & v^k\to v\left(\mathit{strong}\right)\mathit{in}{\mathrm{L}}^2\left({\Omega }_T\right),\mathit{as}k\to 0^{+},\hfill \\ \hfill & {\overline{v}}^k\to v\left(\mathit{strong}\right)\mathit{in}{\mathrm{L}}^2\left({\Omega }_T\right),\mathit{as}k\to 0^{+},\hfill \\ \hfill & \nabla {\overline{v}}^k\rightharpoonup \nabla v\left(\mathit{weak}\right)\mathit{in}{\mathrm{L}}^{\infty }\left(0,T;{\mathrm{L}}^2(\Omega )\right),\mathit{as}k\to 0^{+},\hfill \\ \hfill & v_t^k\rightharpoonup v_t\left(\mathit{weak}\right)\mathit{in}{\mathrm{L}}^{\infty }\left(0,T;{\mathrm{L}}^2(\Omega )\right)\mathit{as}k\to 0^{+}.\hfill \end{array}$$

(79)

Proof. 

Using (50), we obtain that ${\overline{v}}^k$ is uniformly bounded in ${\mathrm{H}}_0^1(\Omega )$, and $\nabla {\overline{v}}^k$ and $v_t^k$ are uniformly bounded in ${\mathrm{L}}^2(\Omega )$. Furthermore, using Lemma 3.2 in [1], we obtain

$${∥v^k(x,t)∥}_{{\mathrm{L}}^2\left({\Omega }_T\right)}^2\le {∥{\overline{v}}^k(x,t)∥}_{{\mathrm{L}}^2\left({\Omega }_T\right)}^2+{\displaystyle \frac{k}{2}}{∥v_0∥}_{{\mathrm{L}}^2(\Omega )}^2$$

(80)

and

$${∥\nabla v^k(x,t)∥}_{{\mathrm{L}}^2\left({\Omega }_T\right)}^2\le {∥\nabla {\overline{v}}^k(x,t)∥}_{{\mathrm{L}}^2\left({\Omega }_T\right)}^2+{\displaystyle \frac{k}{2}}{∥\nabla v_0∥}_{{\mathrm{L}}^2(\Omega )}^2.$$

(81)

Since $v^k$ vanishes on $\partial \Omega$, we find that $v^k$ is uniformly bounded in ${\mathrm{H}}_0^1\left({\Omega }_T\right)$. Those bounds allow us to extract a weakly convergent sub-sequence denoted by $v^k$ again so that we can obtain the following convergence

$$\begin{array}{cc}\hfill & {\overline{v}}^k\to v\left(\mathrm{strong}\right)\mathrm{in}{\mathrm{L}}^2\left({\Omega }_T\right),\mathrm{as}k\to 0^{+},\hfill \\ \hfill & v^k\to u_1\left(\mathrm{strong}\right)\mathrm{in}{\mathrm{L}}^2\left({\Omega }_T\right),\mathrm{as}k\to 0^{+},\hfill \\ \hfill & \nabla {\overline{v}}^k\rightharpoonup u_2\left(\mathrm{weak}\right)\mathrm{in}{\mathrm{L}}^{\infty }\left(0,T;{\mathrm{L}}^2(\Omega )\right),\mathrm{as}k\to 0^{+},\hfill \\ \hfill & v_t^k\rightharpoonup u_3\left(\mathrm{weak}\right)\mathrm{in}{\mathrm{L}}^{\infty }\left(0,T;{\mathrm{L}}^2(\Omega )\right),\mathrm{as}k\to 0^{+}.\hfill \end{array}$$

(82)

By the use of Lemma 3.2 p. 25 in [1], we have that

$${∥v^k-{\overline{v}}^k∥}_{{\mathrm{L}}^2\left({\Omega }_T\right)}^2\le k{∥v_t^k∥}_{{\mathrm{L}}^2\left({\Omega }_T\right)}^2.$$

(83)

Then, both $v^k$ and ${\overline{v}}^k$ have the same limit, i.e., $v=u_1$. Let $\varphi (x,t)\in C_0^{\infty }\left((\Omega \times [0,T))\cap \left\{{\overline{v}}^k>0\right\}\right)$; for all $i=1,\dots ,N$, we have

$${\int }_0^T{\int }_{\Omega }\left({\displaystyle \frac{\partial {\overline{v}}^k(x,t)}{\partial x_i}}-{\displaystyle \frac{\partial v^k(x,t)}{\partial x_i}}\right)\varphi dxdt\to {\int }_0^T{\int }_{\Omega }\left({u_2}_i-{\displaystyle \frac{\partial v(x,t)}{\partial x_i}}\right)\varphi dxdt,\mathrm{as}k\to 0+,$$

(84)

while (83) implies that

$${\int }_0^T{\int }_{\Omega }\left({\displaystyle \frac{\partial {\overline{v}}^k(x,t)}{\partial x_i}}-{\displaystyle \frac{\partial v^k(x,t)}{\partial x_i}}\right)\varphi (x,t)dxdt=-{\int }_0^T{\int }_{\Omega }\left({\overline{v}}^k(x,t)-v^k(x,t)\right){\displaystyle \frac{\partial \varphi (x,t)}{\partial x_i}},dxdt\to 0,$$

(85)

as k → 0+. This leads to the conclusion that $u_2=\nabla v$ almost everywhere in ${\Omega }_T$. Moreover, and in the same manner, we obtain that $u_3=v_t$ almost everywhere in ${\Omega }_T$. □

4. Limit Process

In this section, we will prove that under the assumption $u_0,v_0\in {\mathrm{H}}_0^1(\Omega )$, the approximate weak solutions converge towards a weak solution. We start with problem (1) and problem (3).

Theorem 2.

For $u_0,v_0$ belong to ${\mathrm{H}}_0^1(\Omega )$, then problem (1) and problem (3) have a weak solution.

Proof. 

We take k to approach zero in (45) to reach Definition 1, and we fix an arbitrary $\varphi (x,t)\in C_0^{\infty }(\Omega \times [0,T))$. Because we must utilize a density argument, we approximate $\varphi (x,t)$, the test function, by a series of functions ${\overline{\varphi }}^k\left(x\right)$ that are piecewise constant over the respective partitions of $(0,T)$ and that strongly converge to $\varphi (x,t)$ in ${\mathrm{L}}^2\left(0,T;{\mathrm{H}}^1(\Omega )\right)$. Let us, particularly, set

$${\overline{\varphi }}^k(x,t)={\varphi }_h\left(x\right)={\displaystyle \frac{1}{k}}{\int }_{(m-1)k}^{mk}\varphi (x,t)dt,$$

(86)

for all $(x,t)\in \Omega \times \left(\right(m-1)k,mk],m=1,\dots ,M$. Then, we have

$${\int }_0^T{\int }_{\Omega }{\overline{f}}^k\left(x,t,{\overline{v}}^k\right){\overline{\varphi }}^k(x,t)dxdt={\int }_0^T{\int }_{\Omega }f\left(x,t,{\overline{v}}^k\right){\overline{\varphi }}^k(x,t)dxdt,$$

(87)

and since ${\displaystyle {\displaystyle \frac{v_t^k(x,t)-v_t^k(x,t-k)}{k}}}$ are piecewise constant over the divisions of $(0,T)$, we have

$${\int }_0^T{\int }_{\Omega }{\displaystyle \frac{v_t^k(x,t)-v_t^k(x,t-k)}{k}}{\overline{\varphi }}^k(x,t)dxdt={\int }_0^T{\int }_{\Omega }{\displaystyle \frac{v_t^k(x,t)-v_t^k(x,t-k)}{k}}\varphi (x,t)dxdt.$$

(88)

Thus, (45) becomes

$$\begin{array}{cc}\hfill {\int }_0^T{\int }_{\Omega }& \left({\displaystyle \frac{v_t^k(x,t)-v_t^k(x,t-k)}{k}}\varphi (x,t)+\left(a+b{∥\nabla {\overline{v}}^k(x,t)∥}_{{\mathrm{L}}^2(\Omega )}^2\right)\nabla {\overline{v}}^k(x,t)\nabla {\overline{\varphi }}^k(x,t)\right)dxdt\hfill \\ & +{\int }_0^T{\int }_{\Omega }f\left(x,t,{\overline{v}}^k\right){\overline{\varphi }}^k(x,t)dxdt=0.\hfill \end{array}$$

(89)

We approach the limit as k approaches to 0 in (89), examining the convergence of each term independently. We shall continue in the following manner during the first term:

$$\begin{array}{cc}\hfill & {\int }_0^T{\int }_{\Omega }{\displaystyle \frac{v_t^k(x,t)-v_t^k(x,t-k)}{k}}\varphi (x,t)\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & ={\int }_0^T{\int }_{\Omega }{\displaystyle \frac{v_t^k(x,t)}{k}}\varphi (x,t)\mathrm{d}x\mathrm{d}t-{\int }_0^T{\int }_{\Omega }{\displaystyle \frac{v_t^k(x,t-k)}{k}}\varphi (x,t)\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & ={\int }_0^T{\int }_{\Omega }{v}_t^k(x,t){\displaystyle \frac{\varphi (x,t)}{k}}\mathrm{d}x\mathrm{d}t-{\int }_{-k}^{T-k}{\int }_{\Omega }{v}_t^k(x,t){\displaystyle \frac{\varphi (x,t+k)}{k}}\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & ={\int }_0^T{\int }_{\Omega }{v}_t^k(x,t){\displaystyle \frac{\varphi (x,t)-\varphi (x,t+k)}{k}}\mathrm{d}x\mathrm{d}t-{\int }_{-k}^0{\int }_{\Omega }{\displaystyle \frac{v_t^k(x,t)}{k}}\varphi (x,t+k)\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & +{\int }_{T-k}^T{\int }_{\Omega }{\displaystyle \frac{v_t^k(x,t)}{k}}\varphi (x,t+k)\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & =-{\int }_0^T{\int }_{\Omega }{v}_t^k(x,t){\displaystyle \frac{\varphi (x,t+k)-\varphi (x,t)}{k}}\mathrm{d}x\mathrm{d}t-{\int }_{-k}^0{\int }_{\Omega }{\displaystyle \frac{v_t^k(x,t)}{k}}\varphi (x,t+k)\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & +{\int }_{T-k}^T{\int }_{\Omega }{\displaystyle \frac{v_t^k(x,t)}{k}}\varphi (x,t+k)\mathrm{d}x\mathrm{d}t.\hfill \end{array}$$

(90)

Note that, because $v_t^k(x,t)$ converges weakly to $v_t(x,t)$ in ${\mathrm{L}}^2\left({\Omega }_T\right)$, and ${\displaystyle \frac{\varphi (x,t)-\varphi (x,t+k)}{k}}$ strongly converges to ${\varphi }_t(x,t)$ in ${\mathrm{L}}^2\left({\Omega }_T\right)$, we have that

$${\int }_0^T{\int }_{\Omega }{v}_t^k(x,t){\displaystyle \frac{\varphi (x,t)-\varphi (x,t+k)}{h}}\mathrm{d}x\mathrm{d}t\to {\int }_0^T{\int }_{\Omega }{v}_t(x,t){\varphi }_t(x,t)\mathrm{d}x\mathrm{d}t\mathrm{as}k\to 0^{+},$$

(91)

and due to $v_t^k(x,t)={\displaystyle \frac{(v_1-v_0)}{k}}=u_0\left(x\right)$ for $t\in (-k,0)$, we obtain

$${\int }_{-k}^0{\int }_{\Omega }{\displaystyle \frac{v_t^{k}h(x,t)}{k}}\varphi (x,t+k)\mathrm{d}x\mathrm{d}t\to {\int }_{\Omega }{u}_0\left(x\right)\varphi (0,x)\mathrm{d}x\mathrm{as}k\to 0^{+}.$$

(92)

Furthermore, since $\varphi (x,t+k)=0$ for $(x,t)\in \Omega \times (T-k,T)$ then

$${\int }_{T-k}^T{\int }_{\Omega }{\displaystyle \frac{v_t^k(x,t)}{k}}\varphi (x,t+k)\mathrm{d}x\mathrm{d}t\to 0\mathrm{as}k\to 0^{+},$$

(93)

which leads to

$$\begin{array}{cc}\hfill {\int }_0^T{\int }_{\Omega }{\displaystyle \frac{v_t^k(x,t)-v_t^k(x,t-k)}{k}}\varphi (x,t)\mathrm{d}x\mathrm{d}t\to & -{\int }_0^T{\int }_{\Omega }{v}_t(x,t){\varphi }_t(x,t)\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & -{\int }_{\Omega }{u}_0\left(x\right)\varphi (x,0)\mathrm{d}x\mathrm{as}k\to 0^{+}.\hfill \end{array}$$

(94)

For the outer force term, using the DCT, we obtain

$${\int }_0^T{\int }_{\Omega }f\left(x,t,{\overline{v}}^k\right){\overline{\varphi }}^k(x,t)\mathrm{d}x\mathrm{d}t\to {\int }_0^T{\int }_{\Omega }f\left(x,t,v\right)\varphi (x,t)\mathrm{d}x\mathrm{d}t\mathrm{as}k\to 0^{+}.$$

(95)

For the limit process in the p-Kirchhoff term, we take ${\overline{\varphi }}^k\left(x\right)={\overline{v}}^k(x,t)-v(x,t)$ in (89), and we obtain that

$$\begin{array}{cc}\hfill & {\int }_0^T{\int }_{\Omega }{\displaystyle \frac{v_t^k(x,t)-v_t^k(x,t-k)}{k}}({\overline{v}}^k-v)(x,t)dxdt\hfill \\ \hfill & +{\int }_0^T{\int }_{\Omega }\left(a+b{∥\nabla {\overline{v}}^k(x,t)∥}_{{\mathrm{L}}^2(\Omega )}^2\right)\nabla {\overline{v}}^k(x,t)(\nabla {\overline{v}}^k-\nabla v)(x,t)dxdt\hfill \\ & +{\int }_0^T{\int }_{\Omega }f\left(x,t,{\overline{v}}^k\right)({\overline{v}}^k-v)(x,t)dxdt=0.\hfill \end{array}$$

(96)

By applying the inequality in (60) on the gradient term, we obtain

$$\begin{array}{cc}\hfill -{\int }_0^T& {\int }_{\Omega }{v}_t^k(x,t){\displaystyle \frac{({\overline{v}}^k-v)(x,t+k)-({\overline{v}}^k-v)(x,t)}{k}}\mathrm{d}x\mathrm{d}t-{\int }_{-k}^0{\int }_{\Omega }{\displaystyle \frac{v_t^k(x,t)}{k}}({\overline{v}}^k-v)(x,t+k)\mathrm{d}x\mathrm{d}t\hfill \\ \hfill & +{\int }_{T-k}^T{\int }_{\Omega }{\displaystyle \frac{v_t^k(x,t)}{k}}({\overline{v}}^k-v)(x,t+k)dxdt\hfill \\ & +{\displaystyle \frac{1}{2}}\left(a+b{∥\nabla {\overline{v}}^k(x,t)∥}_{{\mathrm{L}}^2(\Omega )}^2\right){\int }_0^T{\int }_{\Omega }\left({\left|\nabla {\overline{v}}^k(x,t)\right|}^2-{\left|\nabla v(x,t)\right|}^2\right)dxdt\hfill \\ \hfill & +{\int }_0^T{\int }_{\Omega }f\left(x,t,{\overline{v}}^k(x,t)\right)({\overline{v}}^k(x,t)-v(x,t))\mathrm{d}x\mathrm{d}t\le 0.\hfill \end{array}$$

(97)

Taking the limsup as $k\to 0^{+}$, and using the result obtained from (94) and (95), we obtain

$$\underset{k\to 0^{+}}{lim\; sup}{\displaystyle \frac{1}{2}}\left(a+b{∥\nabla {\overline{v}}^k(x,t)∥}_{{\mathrm{L}}^2(\Omega )}^2\right){\int }_0^T{\int }_{\Omega }\left({\left|\nabla {\overline{v}}^k(x,t)\right|}^2-{\left|\nabla v(x,t)\right|}^2\right)dxdt\le 0.$$

(98)

Since $\nabla {\overline{v}}^k$ is bounded in ${\mathrm{L}}^2(\Omega )$, we obtain

$$\underset{k\to 0^{+}}{lim\; sup}{\int }_0^T{\int }_{\Omega }{\left|\nabla {\overline{v}}^k(x,t)\right|}^{2}dxdt\le {\int }_0^T{\int }_{\Omega }{\left|\nabla v(x,t)\right|}^{2}dxdt.$$

(99)

We also have from the WLS-C of the norm in ${\mathrm{L}}^2\left({\Omega }_T\right)$ that

$${\int }_0^T{\int }_{\Omega }{\left|\nabla v(x,t)\right|}^{2}dxdt\le \underset{k\to 0^{+}}{lim\; inf}{\int }_0^T{\int }_{\Omega }{\left|\nabla {\overline{v}}^k(x,t)\right|}^{2}dxdt.$$

(100)

Then, (99) and (100) lead to

$$\underset{k\to 0^{+}}{lim}{\int }_0^T{\int }_{\Omega }{\left|\nabla {\overline{v}}^k(x,t)\right|}^{2}dxdt={\int }_0^T{\int }_{\Omega }{\left|\nabla v(x,t)\right|}^{2}dxdt.$$

(101)

Then, from (94), (101), and (95) together, we find that

$$\begin{array}{cc}\hfill & {\int }_0^T{\int }_{\Omega }\left({\displaystyle \frac{v_t^k(x,t)-v_t^k(x,t-k)}{k}}\varphi (x,t)+\left(a+b{∥\nabla {\overline{v}}^k(x,t)∥}_{{\mathrm{L}}^2(\Omega )}^2\right)\nabla {\overline{v}}^k(x,t)\nabla {\overline{\varphi }}^k\left(x\right)\right)\mathrm{d}x\mathrm{d}t\hfill \\ & +{\int }_0^T{\int }_{\Omega }{\overline{f}}^k\left(x,t,{\overline{v}}^k\right){\overline{\varphi }}^k\left(x\right)\mathrm{d}x\mathrm{d}t\to {\int }_0^T{\int }_{\Omega }\left(-v_t(x,t){\varphi }_t(x,t)+f\left(x,t,v\right)\varphi (x,t)\right)\mathrm{d}x\mathrm{d}t\hfill \\ & +{\int }_0^T{\int }_{\Omega }\left(a+b{∥\nabla v(x,t)∥}_{{\mathrm{L}}^2(\Omega )}^2\right)\nabla v(x,t)\nabla \varphi \mathrm{d}x\mathrm{d}t-{\int }_{\Omega }{u}_0\left(x\right)\varphi (x,0)\mathrm{d}x\mathrm{as}k\to 0^{+}.\hfill \end{array}$$

(102)

which is an identity to Definition 1. Moreover, by construction, $v(x,0)=v_0\left(x\right)$ is fulfilled.

Lastly, we select an arbitrary test function $\phi \in C_0^{\infty }(\Omega \times ([0,T))$ equipped with an arbitrary volume, defining the test function $\varphi$ as

$$\varphi (x,t)=\phi (x,t)-{\displaystyle \frac{v(x,t){\int }_{\Omega }\phi (x,t)\mathrm{d}x}{V}},$$

(103)

then (102) yields an identity to Definition 2:

$$\begin{array}{cc}\hfill {\int }_0^T& {\int }_{\Omega }\left[-v_t(x,t){\phi }_t(x,t)+\left(a+b{∥\nabla v(x,t)∥}_{{\mathrm{L}}^2(\Omega )}^2\right)\nabla v(x,t)\nabla \phi (x,t)+f(x,t,v)\phi (x,t)\right]dxdt\hfill \\ & -{\int }_{\Omega }{u}_0\left(x\right)\phi (x,0)dx=-{\displaystyle \frac{1}{V}}{\int }_{\Omega }{v}_0\left(x\right)u_0\left(x\right)\Phi \left(0\right)dx-{\displaystyle \frac{1}{V}}{\int }_0^T{\int }_{\Omega }{v}_t(x,t){(v\Phi )}_t(x,t)dxdt\hfill \\ & +{\displaystyle \frac{1}{V}}{\int }_0^T{\int }_{\Omega }\left(\left(a+b{∥\nabla v(x,t)∥}_{{\mathrm{L}}^2(\Omega )}^2\right){\left|\nabla v(x,t)\right|}^2+f(x,t,v)v(x,t)\right)\Phi \left(t\right)dxdt,\hfill \end{array}$$

(104)

where

$$\Phi \left(t\right)={\int }_{\Omega }\phi (x,t)dx.$$

(105)

Moreover, by construction, $v(x,0)=v_0\left(x\right)$ is fulfilled. □

Next, for $N=1$, we will prove that under the assumption $u_0,v_0\in {\mathrm{H}}_0^1(\Omega )$, the approximate weak solutions converge towards a weak solution, which proves the existence of a weak solution to problem (2) and problem (4). To achieve this, we first need the following lemma.

Lemma 7.

From the sequence from Lemma 6, there is a subsequence $v^k$ that converges uniformly in ${\Omega }_T$.

Proof. 

For each $t,t_1>0$, we obtain

$$v^k\left(t\right)-v^k\left(t_1\right)={\int }_{t_1}^t{v}_t^k\left(\sigma \right)d\sigma .$$

(106)

The term ${∥v_t^k∥}_{{\mathrm{L}}^{\infty }(0,T;{\mathrm{L}}^2(\Omega ))}$ with respect to k is uniformly bounded. From (106), we obtain

$${∥v^k\left(t\right)-v^k\left(t_1\right)∥}_{{\mathrm{L}}^2(\Omega )}\le {\int }_{t_1}^t{∥v_t^k\left(\sigma \right)∥}_{{\mathrm{L}}^2(\Omega )}d\sigma \le C\left|t-t_1\right|,$$

(107)

where C is an independent constant of k. From here on, C will denote a generic constant independent of k. Moreover, since ${∥v_x^k∥}_{{\mathrm{L}}^{\infty }(0,T;{\mathrm{L}}^2(\Omega ))}$ with respect to k is uniformly bounded, we obtain

$${∥v_x^k\left(t\right)-v_x^k\left(t_1\right)∥}_{{\mathrm{L}}^2(\Omega )}\le {∥v_x^k\left(t\right)∥}_{{\mathrm{L}}^2(\Omega )}+{∥v_x^k\left(t_1\right)∥}_{{\mathrm{L}}^2(\Omega )}\le C.$$

(108)

We obtain this by setting the boundary at the point $x=0$

$$\begin{array}{cc}\hfill {\left(v^k(t,\alpha )-v^k\left(t_1,\alpha \right)\right)}^2=& {\int }_0^{\alpha }{\displaystyle \frac{\partial }{\partial x}}{\left(v^k(t,x)-v^k\left(t_1,x\right)\right)}^{2}dx\hfill \\ \hfill & \le 2{∥v^k\left(t\right)-v^k\left(t_1\right)∥}_{{\mathrm{L}}^2(\Omega )}{∥{\displaystyle \frac{\partial }{\partial x}}\left(v^k\left(t\right)-v^k\left(t_1\right)\right)∥}_{{\mathrm{L}}^2(\Omega )}.\hfill \end{array}$$

(109)

Together with (107) and (108), the above then yields

$${∥v^k\left(t\right)-v^k\left(t_1\right)∥}_{{\mathrm{L}}^{\infty }(\Omega )}\le C{\left|t-t_1\right|}^{{\displaystyle \frac{1}{2}}}.$$

(110)

Then,

$$\begin{array}{cc}\hfill \left|v^k(t,\beta )-v^k(\sigma ,\alpha )\right|& \le \left|v^k(t,\beta )-v^k(t,\alpha )\right|+\left|v^k(t,\alpha )-v^k(\sigma ,\alpha )\right|\hfill \\ & =\left|{\int }_{\alpha }^{\beta }{\displaystyle \frac{\partial }{\partial x}}{v}^k(t,\delta )d\delta \right|+\left|v^k(t,\alpha )-v^k(\sigma ,\alpha )\right|\hfill \\ & \le {\parallel v_x^k\parallel }_{{\mathrm{L}}^{\infty }(0,T;{\mathrm{L}}^2(\Omega ))}{|\beta -\alpha |}^{1/2}+C{|t-\sigma |}^{{\displaystyle \frac{1}{2}}}.\hfill \end{array}$$

(111)

Taking $\sigma =0$ and $\alpha =0$, we find that $v^k$ is uniformly bounded in ${\Omega }_T$. Then, we obtain a subsequence converging uniformly in ${\Omega }_T$ via the Araelà-Ascoli theorem. □

Theorem 3.

For $u_0,v_0$ belonging to ${\mathrm{H}}_0^1(\Omega )$, then problem (2) and problem (4) have weak solutions.

Proof. 

We take k to zero in (45) to reach Definition 1, and we fix an arbitrary test function $\varphi (x,t)\in C_0^{\infty }(\Omega \times [0,T)\cap \{v>0\})$. Due to the continuity of $v(x,t)$ on ${\Omega }_T$, there exists $\nu >0$ where a constant satisfies $v(x,t)\ge \nu$ on $supp\varphi$. The sub-sequence $\left\{v^k(x,t)\right\}$ converges uniformly to $v(x,t)$ by Lemma 6, granting a positive $k_0$ so that

$$\underset{(x,t)\in {\Omega }_T}{max}\left|v^k(x,t)-v(x,t)\right|\le {\displaystyle \frac{\nu }{2}},\forall k<k_0.$$

(112)

From this fact, we obtain that $v^k(x,t)\ge v(x,t)-\left|v^k(x,t)-v(x,t)\right|\ge \nu /2$ on $supp\varphi ,\forall k\in \left(0,k_0\right)$. Since ${\overline{v}}^k(x,t)=v^k(x,mk),\forall (x,t)\in \Omega \times ((m-1)k,mk]$, in addition,

$${\overline{v}}^k(x,t)\ge {\displaystyle \frac{\nu }{2}}>0,\mathrm{on}supp\varphi \forall k\in \left(0,k_0\right).$$

(113)

This means

$$\begin{array}{cc}\hfill {\int }_0^T{\int }_{\Omega }& \left({\displaystyle \frac{v_t^k(x,t)-v_t^k(x,t-k)}{k}}\varphi (x,t)+\left(a+b{∥\nabla {\overline{v}}^k(x,t)∥}_{{\mathrm{L}}^2(\Omega )}^2\right)\nabla {\overline{v}}^k(x,t)\nabla \varphi (x,t)\right)dxdt\hfill \\ & +{\int }_0^T{\int }_{\Omega }{\overline{f}}^k\left(x,t,{\overline{v}}^k\right)\varphi (x,t)dxdt=0\hfill \end{array}$$

(114)

holds to our test function $\varphi (x,t)$, if $k<k_0$. Then, in the same manner as proof of Theorem 2, we obtain that

$$\begin{array}{cc}\hfill & {\int }_0^T{\int }_{\Omega }\left({\displaystyle \frac{v_t^k(x,t)-v_t^k(x,t-k)}{k}}\varphi (x,t)+\left(a+b{∥\nabla {\overline{v}}^k(x,t)∥}_{{\mathrm{L}}^2(\Omega )}^2\right)\nabla {\overline{v}}^k(x,t)\nabla {\overline{\varphi }}^k\left(x\right)\right)\mathrm{d}x\mathrm{d}t\hfill \\ & +{\int }_0^T{\int }_{\Omega }{\overline{f}}^k\left(x,t,{\overline{v}}^k\right){\overline{\varphi }}^k\left(x\right)\mathrm{d}x\mathrm{d}t\to {\int }_0^T{\int }_{\Omega }\left(-v_t(x,t){\varphi }_t(x,t)+f\left(x,t,v\right)\varphi (x,t)\right)\mathrm{d}x\mathrm{d}t\hfill \\ & +{\int }_0^T{\int }_{\Omega }\left(a+b{∥\nabla v(x,t)∥}_{{\mathrm{L}}^2(\Omega )}^2\right)\nabla v(x,t)\nabla \varphi \mathrm{d}x\mathrm{d}t-{\int }_{\Omega }{u}_0\left(x\right)\varphi (x,0)\mathrm{d}x\mathrm{as}k\to 0^{+}.\hfill \end{array}$$

(115)

which is an identity to Definition 3. Moreover, by construction, $v(x,0)=v_0\left(x\right)$ is fulfilled.

Lastly, choosing an arbitrary test function $\phi \in C_0^{\infty }(\Omega \times ([0,T))\cap \{v>0\})$ equipped with volume, we give our test function $\varphi$ the form

$$\varphi (x,t)=\phi (x,t)-{\displaystyle \frac{\tilde{v}(x,t){\int }_{\Omega }\phi (x,t)\mathrm{d}x}{V}},$$

(116)

where $\tilde{v}\in C_0^{\infty }(([0,T]\times \Omega )\cap \{v>0\})$ is an arbitrary function that satisfies ${\int }_{\Omega }\tilde{v}\mathrm{d}x=V$. It is clear that ${\int }_{\Omega }\varphi (t,x)\mathrm{d}x=0$ and $supp\varphi \subset \{v>0\}$. Then, (115) yields an identity similar to the one defined in (4):

$$\begin{array}{cc}\hfill {\int }_0^T& {\int }_{\Omega }\left[-v_t(x,t){\phi }_t(x,t)+\left(a+b{∥\nabla v(x,t)∥}_{{\mathrm{L}}^2(\Omega )}^2\right)\nabla v(x,t)\nabla \phi (x,t)+f(x,t,v)\phi (x,t)\right]dxdt\hfill \\ & -{\int }_{\Omega }{u}_0\left(x\right)\phi (x,0)dx=-{\displaystyle \frac{1}{V}}{\int }_{\Omega }\tilde{v}(x,0)u_0\left(x\right)\Phi \left(0\right)dx-{\displaystyle \frac{1}{V}}{\int }_0^T{\int }_{\Omega }{v}_t(x,t){(\tilde{v}\Phi )}_t(x,t)dxdt\hfill \\ & +{\displaystyle \frac{1}{V}}{\int }_0^T{\int }_{\Omega }\left(\left(a+b{∥\nabla v(x,t)∥}_{{\mathrm{L}}^2(\Omega )}^2\right)\nabla v(x,t)\nabla \tilde{v}(x,t)+f(x,t,v)\tilde{v}(x,t)\right)\Phi \left(t\right)dxdt,\hfill \end{array}$$

(117)

where

$$\Phi \left(t\right)={\int }_{\Omega }\varphi (x,t)dx.$$

(118)

Moreover, by construction, $v(x,t)=0$ outside $\left\{v>0\right\}$, and $v(0,x)=v_0\left(x\right)$ is satisfied. □

5. Conclusions

In this paper, we tackled the intricate problem of establishing the existence of weak solutions for four distinct yet interconnected hyperbolic Kirchhoff-type problems. These include the classical hyperbolic Kirchhoff problem, its extension with a free boundary, a variation incorporating a volume constraint, and a more complex case combining both a volume constraint and a free boundary. Each problem presents unique mathematical challenges, particularly due to the presence of non-local terms introduced by the Kirchhoff term and the interactions of free boundary dynamics with the volume constraint. Despite their theoretical and practical importance, these problems have not been systematically studied before, making our investigation both novel and impactful.

Our approach hinges on the discrete Morse flow (DMF) methodology, which transform continuous-time problems into a sequence of discrete minimization problems. By rigorously constructing approximate weak solutions through the minimization of the functionals ${\mathcal{F}}^i,i=1,\dots ,4$, and refining the time step as $k\to 0$, we demonstrated the convergence of the approximate solutions to a weak solution for each problem. The derivation of energy estimates and the careful passage to the limit were critical to overcoming the difficulties introduced by the interplay of non-local terms and constraints.

To enhance the clarity and readability of this paper, we provide a comparative summary of the four problems and their respective weak formulations. This Table 1 serves as a concise reference for understanding the distinctions and relationships between these problems, highlighting their key characteristics and mathematical structures.

Table 1. Comparison of problems and weak formulations.

Problem	Description	Weak Formulation
Problem (1)	Hyperbolic Kirchhoff-type problem	${\displaystyle -{\int }_0^T{\int }_{\Omega }{v}_t(x,t){\varphi }_t(x,t)dxdt}$
		${\displaystyle +{\int }_0^T{\int }_{\Omega }\left(a+b{∥\nabla v(x,t)∥}_{{\mathrm{L}}^2(\Omega )}^2\right)\nabla v(x,t)\nabla \varphi (x,t)dxdt}$
		${\displaystyle +{\int }_0^T{\int }_{\Omega }f(x,t,v)\varphi (x,t)dxdt-{\int }_{\Omega }{u}_0\left(x\right)\varphi (x,0)dx=0,}$
		for all ${\displaystyle \varphi (x,t)\in C_0^{\infty }((\Omega \times [0,T))}$, and $v(x,0)=v_0$.
Problem (2)	Hyperbolic Kirchhoff-type problem with free boundary	${\displaystyle -{\int }_0^T{\int }_{\Omega }{v}_t(x,t){\varphi }_t(x,t)dxdt}$
		${\displaystyle +{\int }_0^T{\int }_{\Omega }\left(a+b{∥\nabla v(x,t)∥}_{{\mathrm{L}}^2(\Omega )}^2\right)\nabla v(x,t)\nabla \varphi (x,t)dxdt}$
		${\displaystyle +{\int }_0^T{\int }_{\Omega }f(x,t,v)\varphi (x,t)dxdt-{\int }_{\Omega }{u}_0\left(x\right)\varphi (x,0)dx=0,}$
		if $v=0$ outgoing $\left\{v>0\right\}$,$v(x,0)=v_0$, and for all  $\varphi (x,t)\in C_0^{\infty }((\Omega \times [0,T))\cap \left\{v>0\right\})$.
Problem (3)	Hyperbolic Kirchhoff-type problem with volume constraint	${\displaystyle -{\int }_0^T{\int }_{\Omega }{v}_t(x,t){\varphi }_t(x,t)dxdt}$
		${\displaystyle +{\int }_0^T{\int }_{\Omega }\left(a+b{∥\nabla v(x,t)∥}_{{\mathrm{L}}^2(\Omega )}^2\right)\nabla v(x,t)\nabla \varphi (x,t)dxdt}$
		${\displaystyle +{\int }_0^T{\int }_{\Omega }f(x,t,v)\varphi (x,t)dxdt-{\int }_{\Omega }{u}_0\left(x\right)\varphi (x,0)dx=}$
		${\displaystyle -\frac{1}{V}{\int }_{\Omega }{u}_0\left(x\right)v_0\left(x\right)\Phi \left(0\right)dx-\frac{1}{V}{\int }_0^T{\int }_{\Omega }{v}_t(x,t){(v\Phi )}_t(x,t)dxdt}$
		${\displaystyle +\frac{1}{V}{\int }_0^T{\int }_{\Omega }\left(a+b{∥\nabla v(x,t)∥}_{{\mathrm{L}}^2(\Omega )}^2\right){\left|\nabla v(x,t)\right|}^2\Phi \left(t\right)dxdt}$
		${\displaystyle +\frac{1}{V}{\int }_0^T{\int }_{\Omega }f(x,t,v)v(x,t)\Phi \left(t\right)dxdt,}$
		for all $\varphi (x,t)\in C_0^{\infty }((\Omega \times [0,T))$, and $v(x,0)=v_0$, where ${\displaystyle \Phi \left(t\right)={\int }_{\Omega }\varphi (x,t)dx.}$
Problem (4)	Hyperbolic Kirchhoff-type problem with volume constraint and free boundary	${\displaystyle -{\int }_0^T{\int }_{\Omega }{v}_t(x,t){\varphi }_t(x,t)dxdt}$
		${\displaystyle +{\int }_0^T{\int }_{\Omega }\left(a+b{∥\nabla v(x,t)∥}_{{\mathrm{L}}^2(\Omega )}^2\right)\nabla v(x,t)\nabla \varphi (x,t)dxdt}$
		${\displaystyle +{\int }_0^T{\int }_{\Omega }f(x,t,v)\varphi (x,t)dxdt-{\int }_{\Omega }{u}_0\left(x\right)\varphi (x,0)dx=}$
		${\displaystyle -\frac{1}{V}{\int }_{\Omega }\tilde{v}(x,0)v_0\left(x\right)\Phi \left(0\right)dx-\frac{1}{V}{\int }_0^T{\int }_{\Omega }{v}_t(x,t){(\tilde{v}\Phi )}_t(x,t)dxdt}$
		${\displaystyle +\frac{1}{V}{\int }_0^T{\int }_{\Omega }\left(a+b{∥\nabla v(x,t)∥}_{{\mathrm{L}}^2(\Omega )}^2\right)\nabla v(x,t)\nabla \tilde{v}(x,t)\Phi \left(t\right)dxdt}$
		${\displaystyle +\frac{1}{V}{\int }_0^T{\int }_{\Omega }f(x,t,v)\tilde{v}(x,t)\Phi \left(t\right)dxdt,}$
		if $v=0$ outgoing $\left\{v>0\right\}$,$v(x,0)=v_0$, and for all $\varphi (x,t)\in C_0^{\infty }((\Omega \times [0,T))\cap \left\{v>0\right\})$, where
		$\tilde{v}\in C_0^{\infty }(([0,T]\times \Omega )\cap \{v>0\})$ an arbitrary function satisfies ${\displaystyle {\int }_{\Omega }\tilde{v}\mathrm{d}x=V}$, with ${\displaystyle \Phi \left(t\right)={\int }_{\Omega }\varphi (x,t)dx.}$

The novelty of this work lies in its unified and systematic treatment of these complex problems, providing a rigorous framework to handle the interplay of hyperbolic dynamics, free boundary behavior, and volume constraints. To the best of our knowledge, this is the first study to address the existence of weak solutions for all four cases using the DMF approach, bridging gaps in the literature and laying a foundation for further research in this area. The results presented here open new avenues for the study of Kirchhoff-type problems and offer insights that can be applied to a broader class of hyperbolic equations with non-local features.

We believe that our findings contribute significantly to the mathematical understanding of Kirchhoff-type problems and provide a robust framework for tackling similar challenges in future studies.

For completeness, we refer the reader to the Appendix A, where we define important concepts used in the analysis, such as the Sobolev inequality and Grönwall inequality.

Appendix A.1. Hölder’s Inequality

Hölder’s inequality provides an upper bound for the integral or summation of the product of two functions or sequences. For $f,g\in L^p(\Omega )$ and $L^q(\Omega )$, where ${\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}=1$, it states:

$${\int }_{\Omega }\left|f\left(x\right)g\left(x\right)\right|dx\le {\parallel f\parallel }_{L^p(\Omega )}{\parallel g\parallel }_{L^q(\Omega )}.$$

Appendix A.2. Young’s Inequality

Young’s inequality bounds the product of two non-negative numbers using their powers. For $a,b\ge 0$ and $p,q>1$ such that ${\displaystyle \frac{1}{p}}+{\displaystyle \frac{1}{q}}=1$, it states:

$$ab\le {\displaystyle \frac{a^p}{p}}+{\displaystyle \frac{b^q}{q}}.$$

Appendix A.3. Poincaré Inequality

The Poincaré inequality states that for $v\in H_0^1(\Omega )$, there exists a constant $C_P>0$ such that:

$${\parallel v\parallel }_{L^2(\Omega )}\le C_P{\parallel \nabla v\parallel }_{L^2(\Omega )}.$$

This inequality ensures that the $L^2$-norm of a function can be bounded by the norm of its gradient, reflecting the control of a function’s size by its variation.

Appendix A.4. Sobolev Inequality

The Sobolev inequality relates the norms of functions in Sobolev spaces. For $v_h\in H_0^1(\Omega )$, it states:

$$\parallel \nabla v_h{\parallel }_{L^2(\Omega )}\ge C{\parallel v_h\parallel }_{H_0^1(\Omega )},$$

where:

$$\parallel v_h{\parallel }_{H_0^1(\Omega )}^2=\parallel v_h{\parallel }_{L^2(\Omega )}^2+{\parallel \nabla v_h\parallel }_{L^2(\Omega )}^2,$$

and $C>0$ depends only on the domain $\Omega$.

Appendix A.5. Grönwall Inequality

Grönwall’s inequality provides bounds for functions satisfying certain integral inequalities. If $u\left(t\right)$ satisfies

$$u\left(t\right)\le \alpha \left(t\right)+{\int }_{t_0}^t\beta \left(s\right)u\left(s\right)ds,$$

where $\alpha \left(t\right)$ and $\beta \left(t\right)$ are continuous and non-negative, then:

$$u\left(t\right)\le \alpha \left(t\right)+{\int }_{t_0}^t\alpha \left(s\right)\beta \left(s\right)e^{{\int }_s^t\beta \left(\tau \right)d\tau }ds.$$

In a simpler case, with constant $\alpha$ and $\beta$, it simplifies to:

$$u\left(t\right)\le \alpha e^{\beta (t-t_0)}.$$