Analysis of Solutions to Nonlocal Tensor Kirchhoff–Carrier-Type Problems with Strong and Weak Damping, Multiple Mixed Time-Varying Delays, and Logarithmic-Term Forcing

Abstract

In this contribution, we propose and study long-time behaviors of a new class of N-dimensional delayed Kirchhoff–Carrier-type problems with variable transfer coefficients involving a logarithmic nonlinearity. We take into account the dependence of diffusion and damping coefficients on the position and direction, as well as the presence of different types of delays. This class of nonlocal anisotropic and nonlinear wave-type equations with multiple time-varying mixed delays and dampings, of a fairly general form, containing several arbitrary functions and free parameters, is of the following form: ${\displaystyle \frac{{\partial }^{2}u}{\partial t^2}-{div(\mathcal{K}(\Vert \sigma \nabla u\Vert }_{L^2(\Omega )}^2)A_{\sigma }{(\mathbf{x})\nabla u)+\mathcal{M}(\Vert u\Vert }_{L^2(\Omega )}^2)u-div(\zeta (t)A_{\sigma }(\mathbf{x})\nabla \frac{\partial u}{\partial t})+d_0(t)\frac{\partial u}{\partial t}+{\mathbb{D}}_r(\mathbf{x},t;\frac{\partial u}{\partial t})=\mathbb{G}(u),}$ where $u(\mathbf{x},t)$ is the state function, $\mathcal{M}$ and $\mathcal{K}$ are the nonlocal Kirchhoff operators and the nonlinear operator $\mathbb{G}(u)$ corresponds to a logarithmic source term. The symmetric tensor $A_{\sigma }$ describes the anisotropic behavior and processes of the system, and the operator ${\mathbb{D}}_r$ represents the multiple time-varying mixed delays related to velocity ${\displaystyle \frac{\partial u}{\partial t}}$. Our problem, which encompasses numerous equations already studied in the literature, is relevant to a wide range of practical and concrete applications. It not only considers anisotropy in diffusion, but it also assumes that the strong damping can be totally anisotropic (a phenomenon that has received very little mathematical attention in the literature). We begin with the reformulation of the problem into a nonlinear system coupling a nonlocal wave-type equation with ordinary differential equations, with the help of auxiliary functions. Afterward, we study the local existence and some necessary regularity results of the solutions by using the Faedo–Galerkin approximation, combining some energy estimates and the logarithmic Sobolev inequality. Next, by virtue of the potential well method combined with the Nehari manifold, conditions for global in-time existence are given. Finally, subject to certain conditions, the exponential decay of global solutions is established by applying a perturbed energy method. Many of the obtained results can be extended to the case of other nonlinear source terms.

1. Introduction

1.1. Statement of the Problem and Motivation

In the present paper, we introduce and consider the following new class of time-delay Kirchhoff–Carrier-type models involving logarithmic nonlinearity, mixed damping and nonlocal tensors of the general form (for $t>0$)

$$\begin{array}{c}{\displaystyle \frac{{\partial }^{2}u}{\partial t^2}-{div(\mathcal{K}(\Vert \sigma \nabla u\Vert }_{L^2(\Omega )}^2)A_{\sigma }{(\mathbf{x})\nabla u)+\mathcal{M}(\Vert u\Vert }_{L^2(\Omega )}^2)u}\hfill \\ {\displaystyle -div(\zeta (t)A_{\sigma }(\mathbf{x})\nabla \frac{\partial u}{\partial t})+d_0(t)\frac{\partial u}{\partial t}+{\mathbb{D}}_r(\mathbf{x},t;\frac{\partial u}{\partial t})=\mathbb{G}(u),\mathbf{x}\in \Omega ,}\hfill \\ {\displaystyle \frac{\partial u}{\partial t}(\mathbf{x},-t^{\prime })=h_0(\mathbf{x},t^{\prime }),(\mathbf{x},t^{\prime })\in {\mathcal{Q}}_0=\Omega \times (0,\delta (0)),}\hfill \\ {\displaystyle u(\mathbf{x},0)={\varphi }_0(\mathbf{x}),\frac{\partial u}{\partial t}(\mathbf{x},0)={\varphi }_1(\mathbf{x}),\mathbf{x}\in \Omega ,}\hfill \\ {\displaystyle u(\mathbf{x},t)=0,\mathbf{x}\in \partial \Omega ,}\hfill \end{array}$$

(1)

where u is the state function, the nonlocal Kirchhoff operators $\mathcal{M}$ and $\mathcal{K}$ are continuous functions on $[0,+\infty )$, and the domain $\Omega$ is a smoothly bounded domain in ${\mathrm{I}\mathrm{R}}^N$, $N\ge 1$, whose boundary $\Gamma =\partial \Omega$ is suitably regular. The symmetric positive definite matrix function $A_{\sigma }={\sigma }^{*}\sigma$ is the diffusion tensor function on domain $\overline{\Omega }$ ($\overline{\Omega }$ denotes the closure of $\Omega$), where ${\sigma }^{*}\in {\mathrm{I}\mathrm{R}}^{N\times N}$ is the dual of matrix $\sigma \in {\mathrm{I}\mathrm{R}}^{N\times N}$.

If ${\mathcal{K}(\Vert \sigma \nabla u\Vert }_{L^2(\Omega )}^2)A_{\sigma }(\mathbf{x})$ is a scalar, Equation (1) would be called an “isotropic” nonlinear diffusion equation. The application of the diffusion tensor instead of a scalar-valued diffusivity function allows to describe the anisotropic diffusion behavior and processes. The diffusion tensor makes it possible to characterize the magnitude, the degree of anisotropy and the diffusion rate depending on direction. The couple of functions $({\varphi }_0,{\varphi }_1)$ and the function $h_0$ are the initial conditions. The strictly positive bounded parameters $\zeta$ and $d_0$, which are assumed to be sufficiently regular, represent the strong damping and linear weak damping coefficients, respectively. The nonlinear operator $\mathbb{G}$ is defined by the following logarithmic nonlinearity

$$\begin{array}{c}{\displaystyle \mathbb{G}(u)=uln(\mid u{\mid }^{\theta }).}\hfill \end{array}$$

(2)

In this operator $\mathbb{G}$, the non-null real positive parameter $\theta$ plays the role of a measure of strength of the nonlinear interaction.

Finally, the multiple time-varying mixed delays related to velocity ${\displaystyle \frac{\partial u}{\partial t}}$ are represented by ${\mathbb{D}}_r$, which is defined by the following general form (with spatiotemporal integration kernels):

$$\begin{array}{c}{\displaystyle {\mathbb{D}}_r(\mathbf{x},t;\frac{\partial u}{\partial t})=\sum _{j=1}^n{d}_j(t){\int }_{{\alpha }_j}^{{\beta }_j}{e}_j^2(\mathbf{x},\tau )\frac{\partial u}{\partial t}(\mathbf{x},t-c_j(t,\tau ))d\tau ,}\hfill \end{array}$$

(3)

where, for $i\in \left[\right[1,n\left]\right]$, the non-constant weight $e_i$: $\overline{\Omega }\times [{\alpha }_i,{\beta }_i]⟶{\mathrm{I}\mathrm{R}}^{*}$ is a bounded function, with ${\alpha }_i$ and ${\beta }_i$ two real constants satisfying $0\le {\alpha }_i<{\beta }_i$. The bounded function $c_i$ : ${({\mathrm{I}\mathrm{R}}^{+})}^2⟶{\mathrm{I}\mathrm{R}}^{+}$ is a sufficiently regular function representing multiple time-varying delays, and the bounded function $d_i$ : ${\mathrm{I}\mathrm{R}}^{+}⟶\mathrm{I}\mathrm{R}$ is the diffusion coefficient that represents the weight of the time lag associated with it. The real number $\delta (0)$ is defined by ${\displaystyle \delta (0)=\underset{i=1,n}{max}(c_i(0,{\beta }_i))}$.

Furthermore, the operator ${\mathbb{D}}_r$ includes the following situations:

(a)

If $c_i$ is independent on the first variable, for $i\in \left[\right[1,n\left]\right]$, the operator ${\mathbb{D}}_r$ at $(\mathbf{x},t;\varrho (\mathbf{x},t))$ becomes

$$\begin{array}{c}{\displaystyle {\mathbb{D}}_r(\mathbf{x},t;\varrho (\mathbf{x},t))=\sum _{j=1}^n{d}_j(t){\int }_{{\alpha }_j}^{{\beta }_j}{e}_j^2(\mathbf{x},\tau )\varrho (\mathbf{x},t-c_j(\tau ))d\tau .}\hfill \end{array}$$

(4)

Equation (4) corresponds to the case of multiple distributed time-varying delays. Assume that $c_i$ is a strictly increasing function, so there exists an inverse function $b_i$. By a change in variables in the integral, the right-hand side of Equation (4) can also be rewritten as ${\displaystyle \sum _{j=1}^n{d}_j(t){\int }_{t-{\tilde{\beta }}_j}^{t-{\tilde{\alpha }}_j}{k}_j(\mathbf{x},t-s)\varrho (\mathbf{x},s)ds,}$ where $k_j(\mathbf{x},t-s)=b^{\prime }(t-s)e_j^2(\mathbf{x},b(t-s))$, ${\tilde{\alpha }}_j=c_j({\alpha }_j)$ and ${\tilde{\beta }}_j=c_j({\beta }_j)$, and then

$$\begin{array}{c}{\displaystyle {\mathbb{D}}_r(\mathbf{x},t;\varrho (\mathbf{x},t))=\sum _{j=1}^n{d}_j(t){\int }_{t-{\tilde{\beta }}_j}^{t-{\tilde{\alpha }}_j}{k}_j(\mathbf{x},t-s)\varrho (\mathbf{x},s)ds}\hfill \end{array}$$

(5)

This is a particular case of the operator, that was defined in [1].

(b)

If $c_i$ is independent in the second variable, for $i\in \left[\right[1,n\left]\right]$, the operator ${\mathbb{D}}_r$ at $(\mathbf{x},t;\varrho (\mathbf{x},t))$ becomes

$$\begin{array}{c}{\displaystyle {\mathbb{D}}_r(\mathbf{x},t;\varrho (\mathbf{x},t))=\sum _{j=1}^n{\tilde{d}}_j(\mathbf{x},t)\varrho (\mathbf{x},t-c_j(t)).}\hfill \end{array}$$

(6)

where ${\displaystyle {\tilde{d}}_j(\mathbf{x},t)=d_j(t){\int }_{{\alpha }_j}^{{\beta }_j}{e}_j^2(\mathbf{x},\tau )d\tau }$, and then Equation (6) of ${\mathbb{D}}_r$ is defined as in [2].

The motivation for analyzing the dynamical behavior of a class of nonlinear nonlocal anisotropic problems involving general mixed multiple time-varying delays defined in Equation (1), which includes many equations considered in the literature, is due to its wide range of practical and real-world applications in various branches of physics, biology, mechanics, and other theoretical and applied sciences. The introduced nonlinear and nonlocal Kirchhoff–Carrier-type operator with nonlinear anisotropic diffusion:

$${\displaystyle {\mathbb{L}}_{\sigma }(u)=\frac{{\partial }^{2}u}{\partial t^2}-{div(\mathcal{K}(\Vert \sigma \nabla u\Vert }_{L^2(\Omega )}^2)A_{\sigma }{(\mathbf{x})\nabla u)+\mathcal{M}(\Vert u\Vert }_{L^2(\Omega )}^2)u}$$

is a generalization of the classical linear operator ${\displaystyle \mathbb{KG}(u):=\frac{{\partial }^{2}u}{\partial t^2}-\nu \Delta u+m_{0}u}$, with $\nu ,m_0>0$, corresponding to the classical Klein-Gordon equation. Various physical and engineering systems, with wide applications covering different areas such as quantum mechanics, nonlinear optics, condensed matter physics, crystallography, and vibrating systems, are modeled and described mathematically by the Klein–Gordon equation (see, e.g., [3,4]). Nonlinear terms $\mathbb{G}$ and those of ${\mathbb{L}}_{\sigma }$ indicate the presence of complex and nonlinear behaviors that are characteristic of nonlinear systems. The nature of the tensor ${\mathcal{K}(\Vert \sigma \nabla u\Vert }_{L^2(\Omega )}^2)A_{\sigma }(\mathbf{x})$ (dependence on ${\Vert \sigma \nabla u\Vert }_{L^2(\Omega )}^2(t)$) makes it possible to adapt locally and progressively (in time) the tensor during the diffusion process.

The canonical equation of Kirchhoff, later formalized by Carrier, originates from mathematical modeling of small amplitude vibrations of stretched strings, under a suitable approximation (see [5,6]). More precisely, it corresponds to the nonlocal equation ${\displaystyle \frac{{\partial }^{2}u}{\partial t^2}-{\mathcal{K}(\Vert \nabla u\Vert }_{L^2(\Omega )}^2)\Delta u=0}$, where $N=1$, the state variable u is the lateral displacement, and $\mathcal{K}(r)=a+br$ with $a,b>0$. The Kirchhoff–Carrier wave type problem in the isotropic case, with and without damping terms, has been the subject of several studies and provides many interesting results, either from a theoretical (existence and nonexistence of solutions, exponential or polynomial decay of solutions, time blow-up of solutions, etc.) or from a numerical perspective, for different nonlinear source terms $\mathbb{G}$ as power nonlinearity type $\mathbb{G}(u)={\omega }_{f}u\mid u{\mid }^{\gamma -2}$ or logarithmic nonlinearity $\mathbb{G}(u)={\omega }_{f}u\mid u{\mid }^{\gamma -2}ln\mid u\mid$, where ${\omega }_f\in \mathrm{I}\mathrm{R}$ and $\gamma \ge 2$. For power nonlinearities, see, e.g., [7,8,9,10,11,12,13,14] and the references therein. The logarithmic nonlinearity naturally occurs in many areas, such as transport phenomena and quantum optics via a logarithmic Schrodinger model (see, e.g., [15,16]), or such as waves in fluid flows via Korteweg–de Vries model or Kadomtsev–Petviashvili model (see, e.g., [17]). It also occurs naturally in quantum field theory, nuclear physics, inflation cosmology, viscoelastic mechanics, and spinless particles (see, e.g., [18,19,20,21,22,23]). For widely mathematical results, we refer, e.g., to [24,25,26,27,28,29,30,31,32,33,34,35,36] and the references therein.

The introduction of delayed arguments, which are inevitable in various practical and complex systems, aims to reflect the different latencies and aftereffects. Systems with delays represent a different class of infinite-dimensional dynamical systems in which the solution depends on an evolving initial state (including information on the past history). Different time-varying delay configurations (multiple time-varying, distributed, convolution, switching, etc.) naturally appear in various real-world complex systems like biochemical responses, population dynamics, quantum chaotic scattering and relativistic quantum waves. In these delay-sensitive systems, a small delay can significantly affect their dynamical behaviors and performance. In particular, time-lag presence can dramatically influence system stability and response speed, and give rise to a spectrum of complex dynamical behaviors, including instability, poor performance and oscillations (see, e.g., [1,2,24,37,38,39,40,41,42,43,44,45,46,47,48,49] and the references therein). Accordingly, these behaviors and aspects, by considering various sources of delays, make the study and understanding of dynamical systems with mixed time-varying delays crucial.

Classically, the multiplicative damping force on the damped wave equation is considered as an isotropic force. However, in real-world dynamics and under certain conditions, several theoretical and experimental studies have established its anisotropic nature and have suggested expressing it in tensor form. Consequently, the effects of the anisotropy in damping behavior, which is particularly important in magnetic materials (for example, materials with higher spin–orbit coupling strength in different crystallographic directions; see, e.g., [50]), multilayered media, metamaterials, etc., cannot be neglected. Many systems undergoing anisotropic damping forces are studied in materials science and physics (see, e.g., [51,52,53] and the references therein). Accordingly, in order to capture and control the anisotropic damping behavior without introducing material disorder, we have considered and included the time- and space-dependent strong damping with a symmetric anisotropic tensor.

Motivated by the above discussions, to take into consideration the anisotropic diffusion processes, different sources of time lags and different types of damping, we propose and study a new nonlinear and nonlocal Kirchhoff–Carrier-type system by incorporating some general form of mixed multiple time-varying delays, a nonlinear anisotropic diffusion tensor and a linear anisotropic damping tensor. Thus, the derived class of anisotropic, nonlocal and damped history-dependent models (closed by imposing some initial data and the Dirichlet boundary conditions) is given precisely by Equation (1). In the present work, the developed strategy consists of controlling instabilities by imposing some appropriate conditions involving the different functions and parameters representing nonlinear and nonlocal operators, delays and dampings. The nonlocality and nonlinearity of tensor ${\mathcal{K}(\Vert \sigma \nabla u\Vert }_{L^2(\Omega )}^2)A_{\sigma }(\mathbf{x})$ and operator $\mathcal{K}(\Vert u{\Vert }_{L^2(\Omega )}^2)$, the anisotropy of damping and the presence of operator ${\mathbb{D}}_r$, which describes the mixed time-varying delays, cause some mathematical difficulties that make the study of such a class of nonlocal and nonlinear problems particularly challenging.

The rest of the present paper is organized as follows: in Section 1.2, we give some basic definitions and notations and preliminary results useful in the sequel. In Section 2, we start by reformulating the problem into a nonlinear system coupling a nonlocal wave-type equation with ordinary differential equations, with the help of new state functions. Then, we show the local existence of solutions of the derived system using Faedo–Galerkin’s method combined with some energy estimates and logarithmic Sobolev inequality. The Section 3 concerns the potential well and the stability set of the problem. In Section 4, we look into the global existence of solution by virtue of the potential well theory associated with the Nehari manifold. Further, in Section 5, under some appropriate conditions, the exponential decay of the global solution is established. Conclusions are drawn in the final Section 6.

1.2. Assumptions, Notations and Preliminaries

Let $\mathcal{X}$ be a Banach space equipped with the norm ${\Vert .\Vert }_{\mathcal{X}}$ and ${\mathcal{X}}^{\prime }$ (or ${\mathcal{X}}^{*}$) its dual Banach space equipped by the norm ${\Vert .\Vert }_{{\mathcal{X}}^{\prime }}$. The duality pairing of $\mathcal{X}$ with its dual ${\mathcal{X}}^{\prime }$ is given by ${\langle ,\rangle }_{{\mathcal{X}}^{\prime },\mathcal{X}}$ and the inner product in $L^2(\Omega )$ is denoted by $(.,.)$. We say that a sequence ${(g_n)}_n$ of $\mathcal{X}$ (respectively, of ${\mathcal{X}}^{\prime }$) converges weakly (respectively, weakly*) to g if and only if $\forall f\in {\mathcal{X}}^{\prime }$, ${\langle f,g_n-g\rangle }_{{\mathcal{X}}^{\prime },\mathcal{X}}$ converges to 0 (respectively, $\forall v\in \mathcal{X}$, ${\langle g_n-g,v\rangle }_{{\mathcal{X}}^{\prime },\mathcal{X}}$ converges to 0) (see, e.g., [38], Part I). We use the standard notation for Sobolev spaces. The dual of space $L^2(\Omega )$ is identified with itself, and the dual of Sobolev space $H_0^1(\Omega )$ is $H^{-1}(\Omega )$. Moreover, we have the injections $H_0^1(\Omega )\subset L^2(\Omega )\subset H^{-1}(\Omega )$ with continuous and dense embedding, and the following Poincaré inequality

$${\Vert w\Vert }_{L^2(\Omega )}\le C_{\Omega }{\Vert \nabla w\Vert }_{L^2(\Omega )},\forall w\in H_0^1(\Omega ),$$

(7)

where $C_{\Omega }>0$ (depending only upon $\Omega$) is the optimal constant of embedding inequality. This smallest possible $C_{\Omega }$ is named the Poincaré constant and it is equal to ${\displaystyle \frac{d_{\Omega }}{\pi }}$ if the domain $\Omega$ is convex with diameter $d_{\Omega }$. Finally, for a real-valued bounded function g on $[0,+\infty [$, we denote ${\displaystyle g^{max}=\underset{s\ge 0}{sup}g(s),g^{min}=\underset{s\ge 0}{inf}g(s).}$

From now on, we state some assumptions for the various operators, tensors and functions appearing in our model.

We start by assuming the following hypothesis for the time delay functions $c_i$, $i\in \left[\right[1,n\left]\right]$:

Hypothesis 1 (H1).

The function ${({IR}^{+})}^2\ni (t,\tau )⟶c_i(t,\tau )$ is ${\mathcal{C}}^1$ non-negative and bounded function on ${({IR}^{+})}^2$ with ${IR}^{+}\ni \tau ⟶c_i(t,\tau )$ $(\forall t\ge 0)$ and ${IR}^{+}\ni t⟶t-c_i(t,\tau )$ $(\forall \tau \ge 0)$ strictly increasing functions.

We denote by $f_{i0}(\tau )=c_i(0,\tau )$ and $g_{i0}=f_{i0}^{-1}$, $\forall i\in \left[\right[1,n\left]\right]$, and then ${\displaystyle \delta (0)=\underset{i=1,n}{max}{f}_{i0}({\beta }_i)}$. Without loss of generality, one can assume that $\delta (0)=f_{i0}({\beta }_i)$, $\forall i\in \left[\right[1,n\left]\right].$

For the conductivity $\sigma$, we impose the following assumption:

Hypothesis 2 (H2).

We suppose that σ is in $L^{\infty }(\overline{\Omega },{IR}^{N\times N})$ and there exist constants $0<\underline{\varsigma }\le \overline{\varsigma }$ such that (for all $w\in {IR}^N$ and $\mathbf{x}\in \overline{\Omega }$)

$$\begin{array}{c}{\displaystyle \underline{\varsigma }{\parallel w\parallel }_2\le {\parallel \sigma (\mathbf{x})w\parallel }_2\le \overline{\varsigma }{\parallel w\parallel }_2,}\hfill \end{array}$$

(8)

where the notation ${\parallel .\parallel }_2$ denotes the Euclidean norm of a vector.

Remark 1.

(i) For a nonzero vector ${\displaystyle v}$, on $\overline{\Omega }$, the value ${\displaystyle R(A_{\sigma },v)=\frac{{\parallel \sigma v\parallel }_2^2}{{\parallel v\parallel }_2^2}}$ is the Rayleigh quotient for the given matrix ${\displaystyle A_{\sigma }}$. For any vector ${\displaystyle v}$, one has ${\displaystyle R(A_{\sigma },v)\in \left[\underline{\lambda }:=R(A_{\sigma },\underline{v}),\overline{\lambda }:=R(A_{\sigma },\overline{v})\right]}$, where the bounds ${\displaystyle \underline{\lambda }, \overline{\lambda }}$ are, respectively, the smallest and largest eigenvalues of ${\displaystyle A_{\sigma }}$ (with ${\displaystyle \underline{v}, \overline{v}}$ the corresponding eigenvectors, respectively), and that these lower and upper bounds are reached. (ii) The study that we are going to carry out remains the same (with minor modifications) if we replace the strong damping operator, σ, by ${\sigma }_1$, with ${\displaystyle {\underline{\varsigma }}_1{\parallel \sigma v\parallel }_2\le \parallel {\sigma }_1{v\parallel }_2\le {\overline{\varsigma }}_1{\parallel \sigma v\parallel }_2\mathit{in}\overline{\Omega }}$.

The nonlocal operators $\mathcal{M}$ and $\mathcal{K}$, are supposed to satisfy the following assumption:

Hypothesis 3 (H3).

$\mathcal{M}$ and $\mathcal{K}$ are continuous from $[0,\infty )$ into $IR$. Furthermore, the following requirements hold: ${\displaystyle \mathcal{M}(\eta )\ge \frac{{\vartheta }_0}{2}>0}$ and ${\displaystyle \mathcal{K}(\eta )\ge {\vartheta }_1>0}$, $\forall \eta \in [0,\infty )$, with ${\vartheta }_0$ and ${\vartheta }_1$ constants.

We denote, for $s\in [0,\infty )$

$${\displaystyle {\mathcal{M}}_p(s)={\int }_0^s\mathcal{M}(\eta )d\eta \mathit{and}{\mathcal{K}}_p(s)={\int }_0^s\mathcal{K}(\eta )d\eta .}$$

(9)

A simple typical example of Kirchhoff operator $\mathcal{K}$ (resp., $\mathcal{M}$) is given by $\mathcal{K}(\eta )={\vartheta }_1+b_1{\eta }^{{\gamma }_1}$ (resp., $\mathcal{M}(\eta )={\displaystyle \frac{{\vartheta }_0}{2}}+b_0{\eta }^{{\gamma }_0}$), for $\eta \ge 0$, with $b_i>0$ and ${\gamma }_i>0$, for $i=0,1$.

For the strong damping parameter $\zeta$, we suppose that

Hypothesis 4 (H4).

The function $\zeta$ is $C^1$ and bounded on $[0,\infty [$ with $\exists {\zeta }_0\in [0,1[$ such that

$${\zeta }^{\prime }(t)\le 2{\vartheta }_1{\zeta }_0,\forall t\ge 0,$$

where the constant ${\vartheta }_1$ is given in (H3).

Finally, we state the following assumption for value $\theta$: there exists $\alpha >0$ such that

Hypothesis 5 (H5).

(i) ${\displaystyle {\vartheta }_1-\frac{\theta {\alpha }^2}{2\pi {\underline{\varsigma }}^2}>0}$ and (ii) ${\displaystyle {\vartheta }_0+\theta (N(1+ln\alpha )+1)>0}$, where ${\vartheta }_0$ and ${\vartheta }_1$ are constants given in (H3).

From now on, we shall denote by ${\mu }_{\theta }>0$ and ${\kappa }_{\theta }>0$, the following values:

$${\displaystyle {\mu }_{\theta }:={\mu }_{\theta }(\alpha )={\vartheta }_1-\frac{\theta {\alpha }^2}{2\pi {\underline{\varsigma }}^2},{\kappa }_{\theta }:={\kappa }_{\theta }(\alpha )=\frac{{\vartheta }_0+\theta (N(1+ln\alpha )+1)}{2}.}$$

It is clear that the existence of such a parameter $\alpha$ is always guaranteed. Indeed, the condition (i) is valid for any $\alpha$ satisfying ${\displaystyle 0<\alpha <\underline{\varsigma }\sqrt{\frac{2\pi {\vartheta }_1}{\theta }}}$ and relation (ii) is valid when ${\displaystyle exp(-\frac{N+1+\frac{{\vartheta }_0}{\theta }}{N})<\alpha }$. Then, one can always find $\alpha$ satisfying (i) and (ii), provided that the inequality ${\displaystyle \underline{\varsigma }\sqrt{\frac{2\pi {\vartheta }_1}{\theta }}>exp(-\frac{N+1+\frac{{\vartheta }_0}{\theta }}{N})}$ holds, i.e., ${\displaystyle \theta \le {\Theta }_N:={\Theta }_N({\vartheta }_0,{\vartheta }_1,\underline{\varsigma }),}$ with ${\displaystyle {\Theta }_N}$ satisfying the relation ${\displaystyle \underline{\varsigma }\sqrt{\frac{2\pi {\vartheta }_1}{{\Theta }_N}}=exp(-\frac{N+1+\frac{{\vartheta }_0}{{\Theta }_N}}{N})}$. It is clear that ${\displaystyle ({\Theta }_N)}$ is a decreasing sequence and ${\displaystyle {\Theta }_{\infty }=\underset{N\to \infty }{lim}{\Theta }_N=2\pi e^2{\vartheta }_1{\underline{\varsigma }}^2}$. Then, the parameter $\alpha$ satisfies

$${\alpha }_0=exp(-{\displaystyle \frac{N+1+\frac{{\vartheta }_0}{\theta }}{N}})<\alpha <{\alpha }_1=\underline{\varsigma }\sqrt{{\displaystyle \frac{2\pi {\vartheta }_1}{\theta }}}.$$

(10)

Lemma 1.

(Logarithmic Sobolev inequality, [54]). Assume that w is in $H_0^1(\Omega )$ and $\alpha >0$ is any constant. Then,

$$\begin{array}{c}{\displaystyle {\int }_{\Omega }{w}^{2}ln\mid w\mid dx\le \frac{1}{2}{\Vert w\Vert }_{L^2(\Omega )}^{2}ln{\Vert w\Vert }_{L^2(\Omega )}^2}\hfill \\ {\displaystyle +\frac{{\alpha }^2}{2\pi }{\Vert \nabla w\Vert }_{L^2(\Omega )}^2-\frac{N(1+ln\alpha )}{2}{\Vert w\Vert }_{L^2(\Omega )}^2.}\hfill \end{array}$$

(11)

Lemma 2.

(Logarithmic Gronwall inequality, [31]). Let T be a given positive number. Assume that $K_0$ and $g_0$ are positive constants with $g_0\ge 1$, and that the function l is non-negative with $l\in L^1((0,T))$. If $g:(0,T)⟶[1,+\infty [$ satisfies the inequality (for a.e. $t\in (0,T)$)

$$g(t)\le g_0+K_0{\int }_0^{t}l(s)g(s)ln(g(s))ds,$$

then, for a.e. $t\in (0,T)$

$${\displaystyle g(t)\le g_0^{{\displaystyle exp(K_0{\int }_0^{t}l(s)ds)}}.}$$

Definition 1.

Let $T_{max}=sup\{T\in ]0,+\infty ]$: there exists a solution u on $[0,T]$ to problem (1)} be the maximum existence time of u. If $T_{max}=+\infty$ the solution u is said to be global, otherwise u is called a blow-up solution with blow-up time $T_{max}$.

In the rest of the paper, the value C (or $C_i$) will be a generic positive constant.

2. Local Existence of Solution

The purpose of this part is to present the existence of local solutions for Equation (1). Let $T>0$ be a fixed but arbitrary real number and let us denote by $\mathcal{Q}=\Omega \times (0,T)$ and $\Sigma =\Gamma \times (0,T)$.

We start by introducing the following new family of functions ${(w^i)}_{i\in \left[\right[1,n\left]\right]}$:

$${\displaystyle w^i(\mathbf{x},t,\tau ;\eta )=\frac{\partial u}{\partial t}(\mathbf{x},t-\eta c_i(t,\tau ))\mathrm{for}(\mathbf{x},t,\tau ,\eta )\in {\mathbb{T}}_i,}$$

(12)

where ${\mathbb{T}}_i=\mathcal{Q}\times ({\alpha }_i,{\beta }_i)\times (0,1)$.

Then, we have (for $(\mathbf{x},t,\tau ,\eta )\in {\mathbb{T}}_i$, for $i\in \left[\right[1,n\left]\right]$)

$$\begin{array}{c}{\displaystyle w^i(\mathbf{x},t,\tau ;0)=\frac{\partial u}{\partial t}(\mathbf{x},t),w^i(\mathbf{x},t,\tau ;1)=\frac{\partial u}{\partial t}(\mathbf{x},t-c_i(t,\tau )),}\hfill \\ \mathrm{and}\hfill \\ {\displaystyle c_i(t,\tau )\frac{\partial w^i}{\partial t}(\mathbf{x},t,\tau ;\eta )+(1-\eta \frac{\partial c_i}{\partial t}(t,\tau ))\frac{\partial w^i}{\partial \eta }(\mathbf{x},t,\tau ;\eta )=0.}\hfill \end{array}$$

(13)

Therefore, Equation (1) can be written in the form (for $i\in \left[\right[1,n\left]\right]$)

$$\begin{array}{c}{\displaystyle \frac{{\partial }^{2}u}{\partial t^2}-{div(\mathcal{K}(\Vert \sigma \nabla u\Vert }_{L^2(\Omega )}^2)A_{\sigma }{(\mathbf{x})\nabla u)+\mathcal{M}(\Vert u\Vert }_{L^2(\Omega )}^2)u}\hfill \\ {\displaystyle -\zeta (t)div(A_{\sigma }(\mathbf{x})\nabla \frac{\partial u}{\partial t})+d_0(t)\frac{\partial u}{\partial t}}\hfill \\ {\displaystyle +\sum _{j=1}^n{d}_j(t){\int }_{{\alpha }_j}^{{\beta }_j}{e}_j^2(\mathbf{x},\tau )w^j(\mathbf{x},t,\tau ;1)d\tau =\mathbb{G}(u)\mathrm{on}\mathcal{Q},}\hfill \\ {\displaystyle c_i(t,\tau )\frac{\partial w^i}{\partial t}(\mathbf{x},t,\tau ;\eta )+(1-\eta \frac{\partial c_i}{\partial t}(t,\tau ))\frac{\partial w^i}{\partial \eta }(\mathbf{x},t,\tau ;\eta )=0\mathrm{on}{\mathbb{T}}_i,}\hfill \\ {\displaystyle w^i(\mathbf{x},t,\tau ;0)=\frac{\partial u}{\partial t}(\mathbf{x},t)\mathrm{on}\mathcal{Q}\times ({\alpha }_i,{\beta }_i),}\hfill \\ {\displaystyle w^i(\mathbf{x},0,\tau ;\eta )=h_0(\mathbf{x},\eta f_{i0}(\tau )):=w_0^i(\mathbf{x},\tau ,\eta )\mathrm{on}\Omega \times (0,{\beta }_i)\times (0,1),}\hfill \\ {\displaystyle u(\mathbf{x},0)={\varphi }_0(\mathbf{x}),\frac{\partial u}{\partial t}(\mathbf{x},0)={\varphi }_1(\mathbf{x})\mathrm{on}\Omega ,}\hfill \\ {\displaystyle u(\mathbf{x},t)=0\mathrm{on}\Sigma .}\hfill \end{array}$$

(14)

The corresponding energy functional $\mathcal{E}\left[u\right]:{\mathrm{I}\mathrm{R}}^{+}⟶\mathrm{I}\mathrm{R}$, to the state function u, is defined by

$$\begin{array}{c}{\displaystyle \mathcal{E}\left[u\right](t)= \Vert \frac{\partial u}{\partial t}{\Vert }_{L^2(\Omega )}^2+{\mathcal{K}}_p{(\Vert \sigma \nabla u\Vert }_{L^2(\Omega )}^2)+{\mathcal{M}}_p{(\Vert u\Vert }_{L^2(\Omega )}^2)-{\int }_{\Omega }{u}^{2}ln\mid u{\mid }^{\theta }d\mathbf{x}}\hfill \\ {\displaystyle +\frac{\theta }{2}{\Vert u\Vert }_{L^2(\Omega )}^2+\lambda \sum _{j=1}^n{\int }_0^1{\int }_{{\alpha }_j}^{{\beta }_j}{\int }_{\Omega }{c}_j(t,\tau )e_j^2(\mathbf{x},\tau ){(w^j)}^2(\mathbf{x},t,\tau ;\eta )d\mathbf{x}d\tau d\eta ,}\hfill \end{array}$$

(15)

where $\lambda >0$ is a fixed-parameter such that

$$\begin{array}{c}{\displaystyle {{\rm Y}}_1>\lambda >{{\rm Y}}_2\mathrm{with}}\hfill \\ {\displaystyle {{\rm Y}}_1=\underset{(\mathbf{x},t)\in \Omega \times [0,+\infty [}{inf}\frac{{\displaystyle 2d_0(t)-\sum _{j=1}^n\mid d_j(t)\mid {\int }_{{\alpha }_j}^{{\beta }_j}{e}_j^2(\mathbf{x},\tau )d\tau }}{{\displaystyle \sum _{j=1}^n{\int }_{{\alpha }_i}^{{\beta }_i}{e}_j^2(\mathbf{x},\tau )d\tau }},}\hfill \\ {\displaystyle {{\rm Y}}_2=\underset{j=1,n}{max}\left(\underset{(t,\tau )\in [0+\infty )\times ({\alpha }_j,{\beta }_j)}{sup}\mid d_j(t)\mid {(1-\frac{\partial c_j}{\partial t}(t,\tau ))}^{-1}\right)}\hfill \\ \mathrm{under}\mathrm{the}\mathrm{following}\mathrm{additional}\mathrm{assumption}\hfill \\ {\displaystyle {{\rm Y}}_1>{{\rm Y}}_2.}\hfill \end{array}$$

(16)

In the following, we will simply write $\mathcal{E}$ instead of $\mathcal{E}\left[u\right]$, if no confusion arises, and we denote by ${\mathcal{H}}_0({\mathbb{S}}_0)$, where ${\mathbb{S}}_0={\mathcal{Q}}_0\times (0,1)$, the space

$${\displaystyle {\mathcal{H}}_0({\mathbb{S}}_0)=(H_0^1(\Omega )\cap H^2(\Omega ))\times H_0^1(\Omega )\times L^2({\mathbb{S}}_0).}$$

Theorem 1.

Suppose that the assumptions (H1)–(H5) and the condition of Equation (16) hold. Then, for any initial data condition $({\varphi }_0,{\varphi }_1,h_0)\in {\mathcal{H}}_0({\mathbb{S}}_0)$, there exists a local solution u of Equation (1) verifying the regularity ${\displaystyle u\in L^{\infty }(0,T;H_0^1(\Omega )\cap H^2(\Omega ))}$, ${\displaystyle \frac{\partial u}{\partial t}\in L^2(0,T;H_0^1(\Omega ))\cap L^{\infty }(0,T;L^2(\Omega ))}$ and ${\displaystyle \frac{{\partial }^{2}u}{\partial t^2}\in L^{\infty }(0,T;L^2(\Omega ))}$. Moreover, we have that the energy $\mathcal{E}$ is a nonincreasing functional and satisfies the inequality

$$\begin{array}{c}{\displaystyle \frac{d\mathcal{E}}{dt}(t)\le -2{\aleph }_1(\lambda ){\Vert \frac{\partial u}{\partial t}\Vert }_{L^2(\Omega )}^2}\hfill \\ {\displaystyle -2{\aleph }_2(\lambda )\sum _{i=1}^n{\int }_{\Omega }{\int }_{{\alpha }_i}^{{\beta }_i}\mid \frac{\partial u}{\partial t}(\mathbf{x},t-c_i(t,\tau )){\mid }^2{e}_i^2(\mathbf{x},\tau )d\mathbf{x}d\tau \le 0,}\hfill \end{array}$$

(17)

where ${\aleph }_1(\lambda )$ and ${\aleph }_2(\lambda )$ are given by

$$\begin{array}{c}{\displaystyle {\aleph }_1(\lambda )=({{\rm Y}}_1-\lambda )\underset{x\in \Omega }{inf}\sum _{i=1}^n{\int }_{{\alpha }_i}^{{\beta }_i}{e}_i^2(\mathbf{x},\tau )d\tau ,}\hfill \\ {\displaystyle {\aleph }_2(\lambda )=(\lambda -{{\rm Y}}_2).}\hfill \end{array}$$

(18)

Proof. 

To prove the existence of a weak solution to Equation (1), we use the Faedo–Galerkin technique, obtain a priori estimates needed to pass to the limit in approximate solutions and apply compactness methods. Let ${(v_k)}_{k\in \mathrm{I}\mathrm{N}}$ be a complete orthogonal basis of $H_0^1(\Omega )\cap H^2(\Omega )$, which is orthonormal in $L^2(\Omega )$, consisting of the eigenvectors of the operator ${\mathcal{A}}_{\sigma }=div(A_{\sigma }\nabla .)$, which are solutions of homogeneous Dirichlet problem: $-{\mathcal{A}}_{\sigma }{v}_k={\lambda }_k{v}_k$ with $v_k=0$ on $\partial \Omega$, (${(\lambda k)}_{k\in \mathrm{I}\mathrm{N}}$ is the sequence of eigenvalues of the operator $-{\mathcal{A}}_{\sigma }$). For each $m\ge 1$, we denote by $V_m=span\{v_1,\dots ,v_m\}$ the space generated by the finite family of eigenvectors ${(v_k)}_{k=1,m}$. From the family ${(v_k)}_{k=1,m}$, we can derive a sequence ${({\psi }_k^i)}_{k=1,m}$ with ${\psi }_k^i(\mathbf{x},0)=v_k(\mathbf{x})$, for $k=1,m$, such that ${({\psi }_k^i)}_{k=1,m}$ is orthogonal in space $L^2(\Omega \times ({\alpha }_i,{\beta }_i)\times (0,1))$ and then define ${\Psi }_m^i=span\{{\psi }_1^i,\dots ,{\psi }_m^i\}$, for $m\ge 1$. We consider the sequences $u_{m0}$ and $u_{m1}$ in $V_m$, and a sequence $w_{m0}^i$ in ${\Psi }_m^i$, such that $u_{m0}⟶{\varphi }_0$ in $H_0^1(\Omega )\cap H^2(\Omega )$, $u_{m1}⟶{\varphi }_1$ in $H_0^1(\Omega )$ and $w_{m0}^i⟶w_0^i$ in $L^2({\mathcal{Q}}_0\times (0,1))$ as $m\to \infty$. For each $m\in {\mathrm{I}\mathrm{N}}^{*}$, we construct an approximate solution $(u_m,{(w_m^i)}_{i=1,n})$ of Equation (14), in the form

$$u_m(\mathbf{x},t)=\sum _{k=1}^m{h}_{km}(t)v_k(\mathbf{x})\mathrm{and}{w}_m^i(\mathbf{x},t,\tau ;\eta )=\sum _{k=1}^m{g}_{km}^i(t){\psi }_k^i(\mathbf{x},\tau ,\eta ),$$

as follows (a.e. $t\in (0,T)$, $\forall (v,{\psi }^i)\in V_m\times {\Psi }_m^i$ and $i\in \left[\right[1,n\left]\right]$)

$$\begin{array}{c}{\displaystyle \left(\frac{{\partial }^2{u}_m}{\partial t^2},v\right)-\mathcal{K}(\Vert \sigma \nabla u_m{\Vert }_{L^2(\Omega )}^2)\left(div(A_{\sigma }\nabla u_m),v\right)+\left(\mathcal{M}(\Vert u_m{\Vert }_{L^2(\Omega )}^2)u_m,v\right)}\hfill \\ {\displaystyle -\zeta (t)\left(div(A_{\sigma }\nabla \frac{\partial u_m}{\partial t}),v\right)+d_0(t)\left(\frac{\partial u_m}{\partial t},v\right)}\hfill \\ {\displaystyle +\sum _{j=1}^n\left(d_j(t){\int }_{{\alpha }_j}^{{\beta }_j}{e}_j^2(.,\tau )w_m^j(.,t,\tau ;1)d\tau ,v\right)=\left(\mathbb{G}(u_m),v\right),}\hfill \\ {\displaystyle {\int }_0^1\left(c_i(t,\tau )\frac{\partial w_m^i}{\partial t}(.,t,\tau ;\eta ),{\psi }^i(.,\tau ,\eta )\right)d\eta }\hfill \\ {\displaystyle +{\int }_0^1(1-\eta \frac{\partial c_i}{\partial t}(t,\tau ))\left(\frac{\partial w_m^i}{\partial \eta }(.,t,\tau ;\eta ),{\psi }^i(.,\tau ,\eta )\right)d\eta =0,}\hfill \\ {\displaystyle w_m^i(\mathbf{x},t,\tau ;0)=\frac{\partial u_m}{\partial t}(\mathbf{x},t),\mathrm{for}(\mathbf{x},t,\tau )\in \mathcal{Q}\times ({\alpha }_i,{\beta }_i),}\hfill \\ {\displaystyle u_m(\mathbf{x},0)=u_{m0}(\mathbf{x}),\frac{\partial u_m}{\partial t}(\mathbf{x},0)=u_{m1}(\mathbf{x}),\mathrm{for}\mathbf{x}\in \Omega }\hfill \\ {\displaystyle w_m^i(\mathbf{x},0,\tau ;\eta )=w_{m0}^i(\mathbf{x},\tau ,\eta ),\mathrm{for}(\mathbf{x},\tau ,\eta )\in \Omega \times (0,{\beta }_i)\times (0,1)}\hfill \\ {\displaystyle u_m(\mathbf{x},t)=0,(\mathbf{x},t)\in \Sigma .}\hfill \end{array}$$

(19)

So (by using Green’s formula)

$$\begin{array}{c}{\displaystyle \left(\frac{{\partial }^2{u}_m}{\partial t^2},v\right)+\mathcal{K}(\Vert \sigma \nabla u_m{\Vert }_{L^2(\Omega )}^2)\left(\sigma \nabla u_m,\sigma \nabla v\right)+\left(\mathcal{M}(\Vert u_m{\Vert }_{L^2(\Omega )}^2)u_m,v\right)}\hfill \\ {\displaystyle +\zeta (t)\left(\sigma \nabla \frac{\partial u_m}{\partial t},\sigma \nabla v\right)+d_0(t)\left(\frac{\partial u_m}{\partial t},v\right)}\hfill \\ {\displaystyle +\sum _{j=1}^n\left(d_j(t){\int }_{{\alpha }_j}^{{\beta }_j}{e}_j^2(.,\tau )w_m^j(.,t,\tau ;1)d\tau ,v\right)=(\mathbb{G}(u_m),v),}\hfill \\ {\displaystyle {\int }_0^1\left(c_i(t,\tau )\frac{\partial w_m^i}{\partial t}(.,t,\tau ;\eta ),{\psi }^i(.,\tau ,\eta )\right)d\eta }\hfill \\ {\displaystyle +{\int }_0^1(1-\eta \frac{\partial c_i}{\partial t}(t,\tau ))\left(\frac{\partial w_m^i}{\partial \eta }(.,t,\tau ;\eta ),{\psi }^i(.,\tau ,\eta )\right)d\eta =0,}\hfill \\ {\displaystyle w_m^i(\mathbf{x},t,\tau ;0)=\frac{\partial u_m}{\partial t}(\mathbf{x},t),\mathrm{for}(\mathbf{x},t,\tau )\in \mathcal{Q}\times ({\alpha }_i,{\beta }_i),}\hfill \\ {\displaystyle u_m(\mathbf{x},0)=u_{m0}(\mathbf{x}),\frac{\partial u_m}{\partial t}(\mathbf{x},0)=u_{m1}(\mathbf{x}),\mathrm{for}\mathbf{x}\in \Omega ,}\hfill \\ {\displaystyle w_m^i(\mathbf{x},0,\tau ;\eta )=w_{m0}^i(\mathbf{x},\tau ,\eta ),\mathrm{for}(\mathbf{x},\tau ,\eta )\in \Omega \times (0,{\beta }_i)\times (0,1).}\hfill \end{array}$$

(20)

The standard theory of ordinary differential equations guarantees the existence of a solution to Equation (20) in maximal interval $[0,T_m)$, with $0<T_m<T$. Now, we derive a priori estimates for Equation (20), that will give the solution being extended to $[0,T)$.

Unless otherwise specified, the different constants that we will use in the rest of the proof are independent of m.

Now we will establish some necessary a priori estimates. For this, we will need the following three steps:

• Step 1: First a priori estimates

Replacing $(v,{\psi }^i)$ by ${\displaystyle (\frac{\partial u_m}{\partial t},\lambda e_i^2(.,\tau )w_m^i)}$ in Equation (20), using in the first equation of the obtained system, the two following equalities:

$$\begin{array}{c}{\displaystyle \left(\mathbb{G}(u_m),\frac{\partial u_m}{\partial t}\right)=\left(u_{m}ln\mid u_m{\mid }^{\theta },\frac{\partial u_m}{\partial t}\right)}\hfill \\ {\displaystyle ={\int }_{\Omega }\frac{\theta }{2}{u}_{m}ln\mid u_m{\mid }^2\frac{\partial u_m}{\partial t}d\mathbf{x}}\hfill \\ {\displaystyle =\frac{d}{dt}(\frac{1}{2}{\int }_{\Omega }{(u_m)}^{2}ln\mid u_m{\mid }^{\theta }d\mathbf{x}-\frac{\theta }{4}\Vert u_m{\Vert }_{L^2(\Omega )}^2),}\hfill \end{array}$$

and

$$\begin{array}{c}\mathcal{K}(\Vert \sigma \nabla u_m{\Vert }_{L^2(\Omega )}^2)\left(\sigma \nabla u_m,\sigma \nabla {\displaystyle \frac{\partial u_m}{\partial t}}\right)+\left(\mathcal{M}(\Vert u_m{\Vert }_{L^2(\Omega )}^2)u_m,{\displaystyle \frac{\partial u_m}{\partial t}}\right)\hfill \\ {\displaystyle =\frac{1}{2}\mathcal{K}(\Vert \sigma \nabla u_m{\Vert }_{L^2(\Omega )}^2)\frac{d}{dt}\Vert \sigma \nabla u_m{\Vert }_{L^2(\Omega )}^2+\frac{1}{2}\mathcal{M}(\Vert u_m{\Vert }_{L^2(\Omega )}^2)\frac{d}{dt}{\Vert u_m\Vert }_{L^2(\Omega )}^2}\hfill \\ {\displaystyle =\frac{1}{2}\frac{d}{dt}({\mathcal{K}}_p(\Vert \sigma \nabla u_m{\Vert }_{L^2(\Omega )}^2)+{\mathcal{M}}_p(\Vert u_m{\Vert }_{L^2(\Omega )}^2)),(\mathrm{from}\mathrm{the}\mathrm{expression\; (9)})}\hfill \end{array}$$

and, using in the second equation of this system, the following relations (according to the integration by parts in t for the first term and in $\eta$ for the second term):

$$\begin{array}{c}{\displaystyle {\int }_0^1\left(c_i(t,\tau )\frac{\partial w_m^i}{\partial t}(.,t,\tau ;\eta ),e_i^2(.,\tau )w_m^i(.,t,\tau ,\eta )\right)d\eta }\hfill \\ {\displaystyle =\frac{1}{2}{\int }_0^1{c}_i(t,\tau )\frac{\partial }{\partial t}{\Vert e_i(.,\tau )w_m^i(.,t,\tau ;\eta )\Vert }_{L^2(\Omega )}^{2}d\eta }\hfill \\ {\displaystyle =\frac{1}{2}\frac{\partial }{\partial t}\left[c_i(t,\tau ){\int }_0^1{\Vert e_i(.,\tau )w_m^i(.,t,\tau ;\eta )\Vert }_{L^2(\Omega )}^{2}d\eta \right]}\hfill \\ {\displaystyle -\frac{1}{2}\left[\frac{\partial c_i}{\partial t}(t,\tau ){\int }_0^1{\Vert e_i(.,\tau )w_m^i(.,t,\tau ;\eta )\Vert }_{L^2(\Omega )}^{2}d\eta \right]}\hfill \\ {\displaystyle {\int }_0^1(1-\eta \frac{\partial c_i}{\partial t}(t,\tau ))\left(\frac{\partial w_m^i}{\partial \eta }(.,t,\tau ;\eta ),e_i^2(.,\tau )w_m^i(.,\tau ,\eta )\right)d\eta }\hfill \\ {\displaystyle ={\int }_0^1(1-\eta \frac{\partial c_i}{\partial t}(t,\tau ))\frac{\partial }{\partial \eta }{\Vert e_i(.,\tau )w_m^i(.,t,\tau ;\eta )\Vert }_{L^2(\Omega )}^{2}d\eta }\hfill \\ {\displaystyle =\frac{1}{2}\left[\frac{\partial c_i}{\partial t}(t,\tau ){\int }_0^1{\Vert e_i(.,\tau )w_m^i(.,t,\tau ;\eta )\Vert }_{L^2(\Omega )}^{2}d\eta \right]}\hfill \\ {\displaystyle -\frac{1}{2}{\Vert e_i(.,\tau )w_m^i(.,t,\tau ;0)\Vert }_{L^2(\Omega )}^2}\hfill \\ {\displaystyle +\frac{1}{2}\left[\left(1-\frac{\partial c_i}{\partial t}(t,\tau )\right){\Vert e_i(.,\tau )w_m^i(.,t,\tau ;1)\Vert }_{L^2(\Omega )}^2\right]}\hfill \end{array}$$

we get

$$\begin{array}{c}{\displaystyle \frac{1}{2}\frac{d}{dt}[\Vert \frac{\partial u_m}{\partial t}{\Vert }_{L^2(\Omega )}^2+{\mathcal{K}}_p(\Vert \sigma \nabla u_m{\Vert }_{L^2(\Omega )}^2)+{\mathcal{M}}_p(\Vert u_m{\Vert }_{L^2(\Omega )}^2)}\hfill \\ {\displaystyle -{\int }_{\Omega }{(u_m)}^{2}ln\mid u_m{\mid }^{\theta }d\mathbf{x}+\frac{\theta }{2}{\Vert u_m\Vert }_{L^2(\Omega )}^2]}\hfill \\ {\displaystyle +\zeta (t)\Vert \sigma \nabla \frac{\partial u_m}{\partial t}{\Vert }_{L^2(\Omega )}^2+d_0(t){\Vert \frac{\partial u_m}{\partial t}\Vert }_{L^2(\Omega )}^2}\hfill \\ {\displaystyle =-\sum _{j=1}^n{\int }_{\Omega }{\int }_{{\alpha }_j}^{{\beta }_j}{d}_j(t)e_j^2(\mathbf{x},\tau )w_m^j(\mathbf{x},t,\tau ;1)\frac{\partial u_m}{\partial t}(\mathbf{x},t)d\tau d\mathbf{x},}\hfill \\ {\displaystyle \frac{\lambda }{2}\frac{\partial }{\partial t}\left[c_i(t,\tau ){\Vert e_i(.,\tau )w_m^i(.,t,\tau ;.)\Vert }_{L^2(\Omega \times (0,1))}^2\right]=\frac{\lambda }{2}{\Vert e_i(.,\tau )w_m^i(.,t,\tau ;0)\Vert }_{L^2(\Omega )}^2}\hfill \\ {\displaystyle -\frac{\lambda }{2}\left[\left(1-\frac{\partial c_i}{\partial t}(t,\tau )\right){\Vert e_i(.,\tau )w_m^i(.,t,\tau ;1)\Vert }_{L^2(\Omega )}^2\right],}\hfill \\ {\displaystyle w_m^i(\mathbf{x},t,\tau ;0)=\frac{\partial u_m}{\partial t}(\mathbf{x},t),\mathrm{for}(\mathbf{x},t,\tau )\in \mathcal{Q}\times ({\alpha }_i,{\beta }_i),}\hfill \\ {\displaystyle u_m(\mathbf{x},0)=u_{m0}(\mathbf{x}),\frac{\partial u_m}{\partial t}(\mathbf{x},0)=u_{m1}(\mathbf{x}),\mathrm{for}\mathbf{x}\in \Omega }\hfill \\ {\displaystyle w_m^i(\mathbf{x},0,\tau ;\eta )=w_{m0}^i(\mathbf{x},\tau ,\eta ),\mathrm{for}(\mathbf{x},\tau ,\eta )\in \Omega \times (0,{\beta }_i)\times (0,1).}\hfill \end{array}$$

(21)

Let us define the energy ${\mathcal{E}}_m$ for Equation (20) (as in Equation (15))

$$\begin{array}{c}{\displaystyle {\mathcal{E}}_m(t):=\mathcal{E}\left[u_m\right](t)=\Vert \frac{\partial u_m}{\partial t}{\Vert }_{L^2(\Omega )}^2+{\mathcal{K}}_p(\Vert \sigma \nabla u_m{\Vert }_{L^2(\Omega )}^2)+{\mathcal{M}}_p(\Vert u_m{\Vert }_{L^2(\Omega )}^2)}\hfill \\ {\displaystyle -{\int }_{\Omega }{u}_m^{2}ln\mid u_m{\mid }^{\theta }d\mathbf{x}+\frac{\theta }{2}{\Vert u_m\Vert }_{L^2(\Omega )}^2}\hfill \\ {\displaystyle +\lambda \sum _{j=1}^n{\int }_0^1{\int }_{{\alpha }_j}^{{\beta }_j}{\int }_{\Omega }{c}_j(t,\tau )e_j^2(\mathbf{x},\tau ){(w_m^j)}^2(\mathbf{x},t,\tau ;\eta )d\mathbf{x}d\tau d\eta .}\hfill \end{array}$$

(22)

By integrating in $\tau$ the second part of Equation (21) (from ${\alpha }_i$ to ${\beta }_i$), we have for $i\in \left[\right[1,n\left]\right]$ (since $w_m^i(\mathbf{x},t,\tau ;0)={\displaystyle \frac{\partial u_m}{\partial t}}(\mathbf{x},t)$, from the third part of Equation (21))

$$\begin{array}{c}{\displaystyle \frac{\lambda }{2}\frac{d}{dt}\left[{\int }_0^1{\int }_{{\alpha }_i}^{{\beta }_i}{\int }_{\Omega }{c}_i(t,\tau )e_i^2(\mathbf{x},\tau ){(w_m^i)}^2(\mathbf{x},t,\tau ;\eta )d\mathbf{x}d\tau d\eta \right]}\hfill \\ {\displaystyle =\frac{\lambda }{2}{\int }_{\Omega }\left({\int }_{{\alpha }_i}^{{\beta }_i}{e}_i^2(\mathbf{x},\tau )d\tau \right)\mid \frac{\partial u_m}{\partial t}(\mathbf{x},t){\mid }^{2}d\mathbf{x}}\hfill \\ {\displaystyle -\frac{\lambda }{2}{\int }_{\Omega }{\int }_{{\alpha }_i}^{{\beta }_i}\left(1-\frac{\partial c_i}{\partial t}(t,\tau )\right)e_i^2(\mathbf{x},\tau ){(w_m^i)}^2(\mathbf{x},t,\tau ;1)d\tau d\mathbf{x}.}\hfill \end{array}$$

(23)

By summing (for all $1\le i\le n$) Equation (23) and using Equation (22), we can derive the following expression for ${\displaystyle \frac{d{\mathcal{E}}_m}{dt}}$:

$$\begin{array}{c}{\displaystyle \frac{1}{2}\frac{d{\mathcal{E}}_m}{dt}(t)=\frac{1}{2}\frac{d}{dt}[\Vert \frac{\partial u_m}{\partial t}{\Vert }_{L^2(\Omega )}^2+{\mathcal{K}}_p(\Vert \sigma \nabla u_m{\Vert }_{L^2(\Omega )}^2)+{\mathcal{M}}_p(\Vert u_m{\Vert }_{L^2(\Omega )}^2)}\hfill \\ {\displaystyle -{\int }_{\Omega }{u}_m^{2}ln\mid u_m{\mid }^{\theta }d\mathbf{x}+\frac{\theta }{2}{\Vert u_m\Vert }_{L^2(\Omega )}^2]}\hfill \\ {\displaystyle +\frac{\lambda }{2}\sum _{i=1}^n{\int }_{\Omega }\left({\int }_{{\alpha }_i}^{{\beta }_i}{e}_i^2(\mathbf{x},\tau )d\tau \right)\mid \frac{\partial u_m}{\partial t}(\mathbf{x},t){\mid }^{2}d\mathbf{x}}\hfill \\ {\displaystyle -\frac{\lambda }{2}\sum _{i=1}^n{\int }_{\Omega }{\int }_{{\alpha }_i}^{{\beta }_i}\left(1-\frac{\partial c_i}{\partial t}(t,\tau )\right)e_i^2(\mathbf{x},\tau ){(w_m^i)}^2(\mathbf{x},t,\tau ;1)d\tau d\mathbf{x}.}\hfill \end{array}$$

(24)

According to Equation (24), the first part of Equation (21) becomes

$$\begin{array}{c}{\displaystyle \frac{1}{2}\frac{d{\mathcal{E}}_m}{dt}(t)+\zeta (t)\Vert \sigma \nabla \frac{\partial u_m}{\partial t}{\Vert }_{L^2(\Omega )}^2+d_0(t){\Vert \frac{\partial u_m}{\partial t}\Vert }_{L^2(\Omega )}^2}\hfill \\ {\displaystyle =-\sum _{j=1}^n{\int }_{\Omega }{d}_j(t)\left({\int }_{{\alpha }_j}^{{\beta }_j}{e}_j^2(\mathbf{x},\tau )w_m^j(\mathbf{x},t,\tau ;1)\frac{\partial u_m}{\partial t}(\mathbf{x},t)\right)d\tau d\mathbf{x}}\hfill \\ {\displaystyle +\frac{\lambda }{2}\sum _{j=1}^n{\int }_{\Omega }\left({\int }_{{\alpha }_j}^{{\beta }_j}{e}_j^2(\mathbf{x},\tau )d\tau \right)\mid \frac{\partial u_m}{\partial t}(\mathbf{x},t){\mid }^{2}d\mathbf{x}}\hfill \\ {\displaystyle -\frac{\lambda }{2}\sum _{j=1}^n{\int }_{\Omega }{\int }_{{\alpha }_j}^{{\beta }_j}\left(1-\frac{\partial c_j}{\partial t}(t,\tau )\right)e_j^2(\mathbf{x},\tau ){(w_m^j)}^2(\mathbf{x},t,\tau ;1)d\tau d\mathbf{x}.}\hfill \end{array}$$

(25)

According to Young’s inequality, we can deduce

$$\begin{array}{c}{\displaystyle \frac{1}{2}\frac{d{\mathcal{E}}_m}{dt}(t)+\zeta (t)\Vert \sigma \nabla \frac{\partial u_m}{\partial t}{\Vert }_{L^2(\Omega )}^2+d_0(t){\Vert \frac{\partial u_m}{\partial t}\Vert }_{L^2(\Omega )}^2}\hfill \\ {\displaystyle \le \frac{1}{2}\sum _{j=1}^n{\int }_{\Omega }\mid d_j(t)\mid \left({\int }_{{\alpha }_j}^{{\beta }_j}{e}_j^2(\mathbf{x},\tau )d\tau \right)\mid \frac{\partial u_m}{\partial t}(\mathbf{x},t){\mid }^{2}d\mathbf{x}}\hfill \\ {\displaystyle +\frac{1}{2}\sum _{j=1}^n{\int }_{\Omega }\mid d_j(t)\mid \left({\int }_{{\alpha }_j}^{{\beta }_j}{e}_j^2(\mathbf{x},\tau ){(w_m^j)}^2(\mathbf{x},t,\tau ;1)\right)d\tau d\mathbf{x}}\hfill \\ {\displaystyle +\frac{\lambda }{2}\sum _{j=1}^n{\int }_{\Omega }\left({\int }_{{\alpha }_j}^{{\beta }_j}{e}_j^2(\mathbf{x},\tau )d\tau \right)\mid \frac{\partial u_m}{\partial t}(\mathbf{x},t){\mid }^{2}d\mathbf{x}}\hfill \\ {\displaystyle -\frac{\lambda }{2}\sum _{j=1}^n{\int }_{\Omega }{\int }_{{\alpha }_j}^{{\beta }_j}\left(1-\frac{\partial c_j}{\partial t}(t,\tau )\right)e_j^2(\mathbf{x},\tau ){(w_m^j)}^2(\mathbf{x},t,\tau ;1)d\tau d\mathbf{x}.}\hfill \end{array}$$

(26)

This implies

$$\begin{array}{c}{\displaystyle \frac{1}{2}\frac{d{\mathcal{E}}_m}{dt}(t)+\zeta (t){\Vert \sigma \nabla \frac{\partial u_m}{\partial t}\Vert }_{L^2(\Omega )}^2}\hfill \\ {\displaystyle \le -\frac{1}{2}{\int }_{\Omega }\left[2d_0(t)-\sum _{j=1}^n\left(\mid d_j(t)\mid +\lambda \right){\int }_{{\alpha }_j}^{{\beta }_j}{e}_j^2(\mathbf{x},\tau )d\tau \right]\mid \frac{\partial u_m}{\partial t}(\mathbf{x},t){\mid }^{2}d\mathbf{x}}\hfill \\ {\displaystyle -\frac{1}{2}{\int }_{\Omega }\sum _{j=1}^n{\int }_{{\alpha }_j}^{{\beta }_j}\left[\lambda \left(1-\frac{\partial c_j}{\partial t}(t,\tau )\right)-\mid d_j(t)\mid \right]e_j^2(\mathbf{x},\tau ){(w_m^j)}^2(\mathbf{x},t,\tau ;1)d\tau d\mathbf{x}}\hfill \end{array}$$

(27)

and then, according to the condition in Equation (16) on $\lambda$,

$$\begin{array}{c}{\displaystyle \frac{1}{2}\frac{d{\mathcal{E}}_m}{dt}(t)+\zeta (t)\Vert \sigma \nabla \frac{\partial u_m}{\partial t}{\Vert }_{L^2(\Omega )}^2\le -{\aleph }_1(\lambda ){\Vert \frac{\partial u_m}{\partial t}\Vert }_{L^2(\Omega )}^2}\hfill \\ {\displaystyle -{\aleph }_2(\lambda )\sum _{j=1}^n{\int }_{\Omega }{\int }_{{\alpha }_j}^{{\beta }_j}{e}_j^2(\mathbf{x},\tau ){(w_m^j)}^2(\mathbf{x},t,\tau ;1)d\mathbf{x}d\tau }\hfill \\ {\displaystyle \le 0,}\hfill \end{array}$$

(28)

where ${\aleph }_1(\lambda )$ and ${\aleph }_2(\lambda )$ are given by Equation (18).

By integrating Equation (28) over $(0,t)$ ($t\in (0,T)$) and using Equations (8) and (11), we can deduce the following inequality:

$$\begin{array}{c}{\displaystyle \Vert \frac{\partial u_m}{\partial t}{\Vert }_{L^2(\Omega )}^2+\left({\vartheta }_1-\frac{\theta {\alpha }^2}{2\pi {\underline{\varsigma }}^2}\right){\Vert \sigma \nabla u_m\Vert }_{L^2(\Omega )}^2}\hfill \\ {\displaystyle +\frac{\theta (1+N(1+ln\alpha ))+{\vartheta }_0}{2}{\Vert u_m\Vert }_{L^2(\Omega )}^2}\hfill \\ {\displaystyle +{\int }_0^t\zeta (s)\Vert \sigma \nabla \frac{\partial u_m}{\partial t}{(s)\Vert }_{L^2(\Omega )}^{2}ds+2{\aleph }_1(\lambda ){\int }_0^t{\Vert \frac{\partial u_m}{\partial t}(s)\Vert }_{L^2(\Omega )}^{2}ds}\hfill \\ {\displaystyle +2{\aleph }_2(\lambda )\sum _{j=1}^n{\int }_{\Omega }{\int }_0^t{\int }_{{\alpha }_j}^{{\beta }_j}{e}_j^2(\mathbf{x},\tau ){(w_m^j)}^2(\mathbf{x},s,\tau ;1)d\mathbf{x}d\tau ds}\hfill \\ {\displaystyle +\lambda \sum _{j=1}^n{\int }_{\Omega }{\int }_0^1{\int }_{{\alpha }_j}^{{\beta }_j}{c}_j(t,\tau )e_j^2(\mathbf{x},\tau ){(w_m^j)}^2(\mathbf{x},t,\tau ;\eta )d\mathbf{x}d\tau d\eta }\hfill \\ {\displaystyle \le \frac{\theta }{2}\Vert u_m{\Vert }_{L^2(\Omega )}^{2}ln{\Vert u_m\Vert }_{L^2(\Omega )}^2+{\mathcal{E}}_m(0).}\hfill \end{array}$$

(29)

From (H5), we have ${\displaystyle {\mu }_{\theta }={\vartheta }_1-\frac{\theta {\alpha }^2}{2\pi {\underline{\varsigma }}^2}>0,2{\kappa }_{\theta }=\theta (1+N(1+ln\alpha ))+{\vartheta }_0>0}$ and then, from the expression of ${\mathcal{E}}_m(0)$ : ${\mathcal{E}}_m(0)\le C_I({\varphi }_0,{\varphi }_1,h_0)$ (for large m), with $C_I>0$ a constant depending on $({\varphi }_0,{\varphi }_1,h_0)$, we have that

$$\begin{array}{c}{\displaystyle \Vert \frac{\partial u_m}{\partial t}{\Vert }_{L^2(\Omega )}^2+{\mu }_{\theta }\Vert \sigma \nabla u_m{\Vert }_{L^2(\Omega )}^2+{\kappa }_{\theta }{\Vert u_m\Vert }_{L^2(\Omega )}^2}\hfill \\ {\displaystyle +{\int }_0^t\zeta (s)\Vert \sigma \nabla \frac{\partial u_m}{\partial t}{(s)\Vert }_{L^2(\Omega )}^{2}ds+2{\aleph }_1(\lambda ){\int }_0^t{\Vert \frac{\partial u_m}{\partial t}(s)\Vert }_{L^2(\Omega )}^{2}ds}\hfill \\ {\displaystyle +2{\aleph }_2(\lambda )\sum _{j=1}^n{\int }_{\Omega }{\int }_0^t{\int }_{{\alpha }_j}^{{\beta }_j}{e}_j^2(\mathbf{x},\tau ){(w_m^j)}^2(\mathbf{x},s,\tau ;1)d\mathbf{x}d\tau ds}\hfill \\ {\displaystyle +\lambda \sum _{j=1}^n{\int }_{\Omega }{\int }_0^1{\int }_{{\alpha }_j}^{{\beta }_j}{c}_j(t,\tau )e_j^2(\mathbf{x},\tau ){(w_m^j)}^2(\mathbf{x},t,\tau ;\eta )d\mathbf{x}d\tau d\eta }\hfill \\ {\displaystyle \le C_I+\frac{\theta }{2}\Vert u_m{\Vert }_{L^2(\Omega )}^{2}ln{\Vert u_m\Vert }_{L^2(\Omega )}^2.}\hfill \end{array}$$

(30)

As ${\displaystyle u_m(.,t)=u_m(.,0)+{\int }_0^t\frac{\partial u_m}{\partial t}(.,s)ds}$, then, for large m, (according to Equation (30))

$$\begin{array}{c}{\displaystyle \Vert u_m{(t)\Vert }_{L^2(\Omega )}^2\le 2\Vert u_m{(0)\Vert }_{L^2(\Omega )}^2+C_1{\int }_0^t{\Vert \frac{\partial u_m}{\partial t}(s)\Vert }_{L^2(\Omega )}^{2}ds}\hfill \\ {\displaystyle \le C_0+C_2{\int }_0^t\Vert u_m{(s)\Vert }_{L^2(\Omega )}^{2}ln{\Vert u_m(s)\Vert }_{L^2(\Omega )}^{2}ds.}\hfill \end{array}$$

(31)

From logarithmic Gronwall inequality, one has the following estimate (for a.e. $t\in (0,T)$):

$$\Vert u_m{(t)\Vert }_{L^2(\Omega )}^2\le C_T.$$

(32)

Then from Equation (30), according to Equation (32) and the boundedness of $\zeta$, it follows (for a.e. $t\in (0,T)$):

$$\begin{array}{c}{\displaystyle \Vert \frac{\partial u_m}{\partial t}{\Vert }_{L^2(\Omega )}^2+\Vert \sigma \nabla u_m{\Vert }_{L^2(\Omega )}^2+\Vert u_m{\Vert }_{L^2(\Omega )}^2+{\int }_0^t{\Vert \sigma \nabla \frac{\partial u_m}{\partial t}(s)\Vert }_{L^2(\Omega )}^{2}ds}\hfill \\ {\displaystyle +{\int }_0^t{\Vert \frac{\partial u_m}{\partial t}(s)\Vert }_{L^2(\Omega )}^{2}ds+\sum _{j=1}^n{\int }_{\Omega }{\int }_0^t{\int }_{{\alpha }_j}^{{\beta }_j}{e}_j^2(\mathbf{x},\tau ){(w_m^j)}^2(\mathbf{x},s,\tau ;1)d\mathbf{x}d\tau ds}\hfill \\ {\displaystyle +\sum _{j=1}^n{\int }_{\Omega }{\int }_0^1{\int }_{{\alpha }_j}^{{\beta }_j}{c}_j(t,\tau )e_j^2(\mathbf{x},\tau ){(w_m^j)}^2(\mathbf{x},t,\tau ;\eta )d\mathbf{x}d\tau d\eta \le C.}\hfill \end{array}$$

(33)

This ensures that (for all $1\le j\le n$)

$$\begin{array}{c}{\displaystyle (u_m)\mathrm{is}\mathrm{bounded}\mathrm{in}{L}^{\infty }(0,T;H_0^1(\Omega )),}\hfill \\ {\displaystyle (\frac{\partial u_m}{\partial t})\mathrm{is}\mathrm{bounded}\mathrm{in}{L}^{\infty }(0,T;L^2(\Omega ))\cap L^2(0,T;H_0^1(\Omega )),}\hfill \\ {\displaystyle (e_j{w}_m^j(.,.;1))\mathrm{is}\mathrm{bounded}\mathrm{in}{L}^2(0,T;L^2(\Omega \times ({\alpha }_i,{\beta }_i))),}\hfill \\ {\displaystyle (\sqrt{c_j}{e}_j{w}_m^j)\mathrm{is}\mathrm{bounded}\mathrm{in}{L}^{\infty }(0,T;L^2(\Omega \times ({\alpha }_i,{\beta }_i)\times (0,1))).}\hfill \end{array}$$

(34)

Moreover, since ${\displaystyle \underset{0<y\le 1}{sup}\mid ylny\mid =e^{-1}}$, we get that

$$\begin{array}{c}{\displaystyle \Vert u_{m}ln\mid u_m{\mid }^{\theta }{\Vert }_{L^2(\Omega )}^2\le C_1\left[1+\Vert u_m{\Vert }_{L^q(\Omega )}^q\right]\le C\left[1+\Vert \sigma \nabla u_m{\Vert }_{L^2(\Omega )}^q\right],}\hfill \end{array}$$

(35)

with $\forall q>2$ if $N\le 2$ and $2<q\le {\displaystyle \frac{2N}{N-2}}$ if $N\ge 3$. Then, from the first relation of Equation (34), and Equation (35),

$${\displaystyle (u_{m}ln\mid u_m{\mid }^{\theta })\mathrm{is}\mathrm{bounded}\mathrm{in}{L}^{\infty }(0,T;L^2(\Omega )).}$$

(36)

• Step 2: Second a priori estimates

Replacing v by ${\displaystyle -{\mathcal{A}}_{\sigma }{u}_m}$ in the first part of Equation (19), where ${\displaystyle {\mathcal{A}}_{\sigma }.=div(A_{\sigma }\nabla .)}$, we obtain

$$\begin{array}{c}{\displaystyle \frac{d}{dt}\left[\left(\frac{\partial u_m}{\partial t},-{\mathcal{A}}_{\sigma }{u}_m\right)+\frac{\zeta (t)}{2}{\Vert {\mathcal{A}}_{\sigma }{u}_m\Vert }_{L^2(\Omega )}^2\right]-{\Vert \sigma \nabla \frac{\partial u_m}{\partial t}\Vert }_{L^2(\Omega )}^2}\hfill \\ {\displaystyle +\mathcal{K}(\Vert \sigma \nabla u_m{\Vert }_{L^2(\Omega )}^2)\Vert {\mathcal{A}}_{\sigma }{u}_m{\Vert }_{L^2(\Omega )}^2+\mathcal{M}(\Vert u_m{\Vert }_{L^2(\Omega )}^2)\Vert \sigma \nabla u_m{\Vert }_{L^2(\Omega )}^2}\hfill \\ {\displaystyle +d_0(t)\left(\frac{\partial u_m}{\partial t},-{\mathcal{A}}_{\sigma }{u}_m\right)-\frac{{\zeta }^{\prime }(t)}{2}{\Vert {\mathcal{A}}_{\sigma }{u}_m\Vert }_{L^2(\Omega )}^2}\hfill \\ {\displaystyle +\sum _{j=1}^n\left(d_j(t){\int }_{{\alpha }_j}^{{\beta }_j}{e}_j^2(.,\tau )w_m^j(.,t,\tau ;1)d\tau ,-{\mathcal{A}}_{\sigma }{u}_m\right)}\hfill \\ {\displaystyle =\left(u_{m}ln\mid u_m{\mid }^{\theta },-{\mathcal{A}}_{\sigma }{u}_m\right).}\hfill \end{array}$$

(37)

Then from (H3), Equation (35) and Young’s inequality (according to the boundedness of functions $d_0$, $d_j$ and $e_j$, for $j=0,n$)

$$\begin{array}{c}{\displaystyle \frac{d}{dt}\left[\left(\frac{\partial u_m}{\partial t},-{\mathcal{A}}_{\sigma }{u}_m\right)+\frac{\zeta (t)}{2}{\Vert {\mathcal{A}}_{\sigma }{u}_m\Vert }_{L^2(\Omega )}^2\right]+{\vartheta }_1\Vert {\mathcal{A}}_{\sigma }{u}_m{\Vert }_{L^2(\Omega )}^2+\frac{{\vartheta }_0}{2}{\Vert \sigma \nabla u_m\Vert }_{L^2(\Omega )}^2}\hfill \\ {\displaystyle \le \Vert d_0{\Vert }_{\infty }\Vert \frac{\partial u_m}{\partial t}{\Vert }_{L^2(\Omega )}\Vert {\mathcal{A}}_{\sigma }{u}_m{\Vert }_{L^2(\Omega )}+\frac{{\zeta }^{\prime }(t)}{2}{\Vert {\mathcal{A}}_{\sigma }{u}_m\Vert }_{L^2(\Omega )}^2}\hfill \\ {\displaystyle +\sum _{j=1}^n\Vert d_j{\Vert }_{\infty }\Vert e_j{\Vert }_{L^{\infty }(\Omega \times ({\alpha }_j,{\beta }_j))}{\left({\int }_{\Omega }{\int }_{{\alpha }_j}^{{\beta }_j}{e}_j^2(\mathbf{x},\tau ){(w_m^j)}^2(\mathbf{x},t,\tau ;1)d\tau d\mathbf{x}\right)}^{1/2}{\Vert {\mathcal{A}}_{\sigma }{u}_m\Vert }_{L^2(\Omega )}}\hfill \\ {\displaystyle +\Vert u_{m}ln\mid u_m{\mid }^{\theta }{\Vert }_{L^2(\Omega )}\Vert {\mathcal{A}}_{\sigma }{u}_m{\Vert }_{L^2(\Omega )}+{\Vert \sigma \nabla \frac{\partial u_m}{\partial t}\Vert }_{L^2(\Omega )}^2}\hfill \\ {\displaystyle \le C_1\left[1+\Vert \frac{\partial u_m}{\partial t}{\Vert }_{L^2(\Omega )}^2+{\Vert \sigma \nabla u_m\Vert }_{L^2(\Omega )}^q+\sum _{j=1}^n{\int }_{\Omega }{\int }_{{\alpha }_j}^{{\beta }_j}{e}_j^2(\mathbf{x},\tau ){(w_m^j)}^2(\mathbf{x},t,\tau ;1)d\tau d\mathbf{x}\right]}\hfill \\ {\displaystyle +\left({ϵ}_0+\frac{{\zeta }^{\prime }(t)}{2}\right)\Vert {\mathcal{A}}_{\sigma }{u}_m{\Vert }_{L^2(\Omega )}^2+{\Vert \sigma \nabla \frac{\partial u_m}{\partial t}\Vert }_{L^2(\Omega )}^2,}\hfill \end{array}$$

where the value of ${ϵ}_0>0$ will be chosen appropriately.

From Equation (34) and assumption (H4), it follows that (by taking ${ϵ}_0\le {\vartheta }_1(1-{\zeta }_0)$):

$$\begin{array}{c}{\displaystyle \frac{d}{dt}\left[\left(\frac{\partial u_m}{\partial t},-{\mathcal{A}}_{\sigma }{u}_m\right)+\frac{\zeta (t)}{2}{\Vert {\mathcal{A}}_{\sigma }{u}_m\Vert }_{L^2(\Omega )}^2\right]+\frac{{\vartheta }_0}{2}{\Vert \sigma \nabla u_m\Vert }_{L^2(\Omega )}^2}\hfill \\ {\displaystyle \le C_2\left[1+\sum _{j=1}^n{\int }_{\Omega }{\int }_{{\alpha }_j}^{{\beta }_j}{e}_j^2(\mathbf{x},\tau ){(w_m^j)}^2(\mathbf{x},t,\tau ;1)d\tau d\mathbf{x}\right]+{\Vert \sigma \nabla \frac{\partial u_m}{\partial t}\Vert }_{L^2(\Omega )}^2.}\hfill \end{array}$$

(38)

By integrating Equation (38) from 0 to t, we have that (according to Equation (34) and the regularity of initial conditions)

$$\begin{array}{c}{\displaystyle \frac{{inf}_{t\ge 0}\zeta (t)}{4}\Vert {\mathcal{A}}_{\sigma }{u}_m{\Vert }_{L^2(\Omega )}^2+\frac{{\vartheta }_0}{2}{\int }_0^t{\Vert \sigma \nabla u_m\Vert }_{L^2(\Omega )}^{2}ds}\hfill \\ {\displaystyle \le C_3\left[1+\Vert u_{1m}{\Vert }_{L^2(\Omega )}^2+{\Vert {\mathcal{A}}_{\sigma }{u}_{0m}\Vert }_{L^2(\Omega )}^2\right]\le C_4.}\hfill \end{array}$$

(39)

This implies

$$u_m\mathrm{is}\mathrm{bounded}\mathrm{in}{L}^{\infty }(0,T;H_0^1(\Omega )\cap H^2(\Omega )).$$

(40)

• Step 3: Third a priori estimates

Replacing v by ${\displaystyle \frac{{\partial }^2{u}_m}{\partial t^2}}$ in the first part of Equation (19), we obtain

$$\begin{array}{c}{\displaystyle \Vert \frac{{\partial }^2{u}_m}{\partial t^2}{\Vert }_{L^2(\Omega )}^2+\frac{1}{2}\frac{d}{dt}\left[\zeta (t)\Vert \sigma \nabla \frac{\partial u_m}{\partial t}{\Vert }_{L^2(\Omega )}^2\right]}\hfill \\ {\displaystyle =\mathcal{K}(\Vert \sigma \nabla u_m{\Vert }_{L^2(\Omega )}^2)\left({\mathcal{A}}_{\sigma }{u}_m,\frac{{\partial }^2{u}_m}{\partial t^2}\right)-\mathcal{M}(\Vert u_m{\Vert }_{L^2(\Omega )}^2)\left(u_m,\frac{{\partial }^2{u}_m}{\partial t^2}\right)}\hfill \\ {\displaystyle +\frac{{\zeta }^{\prime }(t)}{2}{\Vert \sigma \nabla \frac{\partial u_m}{\partial t}\Vert }_{L^2(\Omega )}^2-d_0(t)\left(\frac{\partial u_m}{\partial t},\frac{{\partial }^2{u}_m}{\partial t^2}\right)}\hfill \\ {\displaystyle -\sum _{j=1}^n\left(d_j(t){\int }_{{\alpha }_j}^{{\beta }_j}{e}_j^2(.,\tau )w_m^j(.,t,\tau ;1)d\tau ,\frac{{\partial }^2{u}_m}{\partial t^2}\right)+\left(u_{m}ln\mid u_m{\mid }^{\theta },\frac{{\partial }^2{u}_m}{\partial t^2}\right).}\hfill \end{array}$$

(41)

According to Equations (34) and (40) and the continuity of $\mathcal{K}$ and $\mathcal{M}$, we get

$$\begin{array}{c}{\displaystyle \mathcal{K}(\Vert \sigma \nabla u_m{\Vert }_{L^2(\Omega )}^2)\left({\mathcal{A}}_{\sigma }{u}_m,\frac{{\partial }^2{u}_m}{\partial t^2}\right)-\mathcal{M}(\Vert u_m{\Vert }_{L^2(\Omega )}^2)\left(u_m,\frac{{\partial }^2{u}_m}{\partial t^2}\right)}\hfill \\ {\displaystyle \le C_5+{ϵ}_1{\Vert \frac{{\partial }^2{u}_m}{\partial t^2}\Vert }_{L^2(\Omega )}^2,}\hfill \end{array}$$

where the value of ${ϵ}_1>0$ will be chosen appropriately later. Then (according to Equation (34), assumption (H4) and estimate of Equation (35))

$$\begin{array}{c}{\displaystyle \Vert \frac{{\partial }^2{u}_m}{\partial t^2}{\Vert }_{L^2(\Omega )}^2+\frac{1}{2}\frac{d}{dt}\left[\zeta (t)\Vert \sigma \nabla \frac{\partial u_m}{\partial t}{\Vert }_{L^2(\Omega )}^2\right]}\hfill \\ {\displaystyle \le C_6\left[1+\sum _{j=1}^n{\int }_{\Omega }{\int }_{{\alpha }_j}^{{\beta }_j}{e}_j^2(\mathbf{x},\tau ){(w_m^j)}^2(\mathbf{x},t,\tau ;1)d\tau d\mathbf{x}+{\Vert \sigma \nabla \frac{\partial u_m}{\partial t}\Vert }_{L^2(\Omega )}^2\right]}\hfill \\ {\displaystyle +\left({ϵ}_1+{ϵ}_2\right){\Vert \frac{{\partial }^2{u}_m}{\partial t^2}\Vert }_{L^2(\Omega )}^2,}\hfill \end{array}$$

(42)

where the value of ${ϵ}_2>0$ will be chosen appropriately.

By choosing ${ϵ}_1$ and ${ϵ}_0$ such that ${ϵ}_1+{ϵ}_0=1/2$ and by integrating Equation (42) from 0 to t, we can deduce that (according to Equation (34) and the regularity of the initial conditions)

$$\begin{array}{c}{\displaystyle {\int }_0^t\Vert \frac{{\partial }^2{u}_m}{\partial t^2}{\Vert }_{L^2(\Omega )}^{2}ds+\zeta (t){\Vert \sigma \nabla \frac{\partial u_m}{\partial t}\Vert }_{L^2(\Omega )}^2\le C_7.}\hfill \end{array}$$

(43)

Consequently

$$\begin{array}{c}{\displaystyle \frac{{\partial }^2{u}_m}{\partial t^2}\mathrm{is}\mathrm{bounded}\mathrm{in}{L}^2(0,T;L^2(\Omega ))}\hfill \\ {\displaystyle \frac{\partial u_m}{\partial t}\mathrm{is}\mathrm{bounded}\mathrm{in}{L}^{\infty }(0,T;H_0^1(\Omega )).}\hfill \end{array}$$

(44)

We can now perform the limit process as $m\to \infty$ and show that, in the limit, we derive the weak solution.

By the weak compactness theorem, it follows from the a priori estimates of Equations (34), (36), (40) and (44) that there exists a subsequence of $(u_m,w_m^i)$ also denoted by $(u_m,w_m^i)$ such that the following weak convergence results hold true as $m\to \infty$:

$$\begin{array}{c}{\displaystyle u_m\stackrel{*}{\rightharpoonup }u\mathrm{weakly}*\mathrm{in}{L}^{\infty }(0,T;H_0^1(\Omega )\cap H^2(\Omega )),}\hfill \\ {\displaystyle \frac{\partial u_m}{\partial t}\stackrel{*}{\rightharpoonup }\frac{\partial u}{\partial t}\mathrm{weakly}*\mathrm{in}{L}^{\infty }(0,T;H_0^1(\Omega )),}\hfill \\ {\displaystyle \frac{{\partial }^2{u}_m}{\partial t^2}\rightharpoonup \frac{{\partial }^{2}u}{\partial t^2}\mathrm{weakly}\mathrm{in}{L}^2(0,T;L^2(\Omega )),}\hfill \\ {\displaystyle \sqrt{c_i}{e}_i{w}_m^i\stackrel{*}{\rightharpoonup }\sqrt{c_i}{e}_i{w}^i\mathrm{weakly}*\mathrm{in}{L}^{\infty }(0,T;L^2(\Omega \times ({\alpha }_i,{\beta }_i)\times (0,1))),}\hfill \\ {\displaystyle e_i{w}_m^i(.,.;1)\rightharpoonup e_i{w}^i(.,.;1)\mathrm{weakly}\mathrm{in}{L}^2(0,T;L^2(\Omega \times ({\alpha }_i,{\beta }_i))).}\hfill \end{array}$$

(45)

In addition, it follows in particular from Equation (40) and the second estimate of Equation (44) that $u_m$ is bounded in $L^2(0,T;H_0^1(\Omega )\cap H^2(\Omega ))$ and ${\displaystyle \frac{\partial u_m}{\partial t}}$ is bounded in $L^2(0,T;H_0^1(\Omega ))$. Then, by using Aubin–Lions’s lemma [55], we can deduce that there exists a subsequence of $u_m$, still denoted by $u_m$, such that

$$\begin{array}{c}{\displaystyle u_m⟶u\mathrm{strongly}\mathrm{in}{L}^2(0,T;H_0^1(\Omega ))\mathrm{and}\mathrm{so}}\hfill \\ {\displaystyle u_m⟶u,\sigma \nabla u_m⟶\sigma \nabla u\mathrm{a}.\mathrm{e}.\mathrm{in}\mathcal{Q}.}\hfill \end{array}$$

(46)

Moreover, from the continuity of $\mathcal{K}$, $\mathcal{M}$ and of the function $y⟶yln\mid y{\mid }^{\theta }$ (using Equation (46)), we can deduce the following convergence results:

$$\begin{array}{c}\mathcal{K}(\Vert \sigma \nabla u_m{\Vert }_{L^2(\Omega )}^2{)⟶\mathcal{K}(\Vert \sigma \nabla u\Vert }_{L^2(\Omega )}^2)\mathrm{strongly}\mathrm{in}{L}^2(0,T)\mathrm{and}\hfill \\ \mathcal{M}(\Vert u_m{\Vert }_{L^2(\Omega )}^2{)⟶\mathcal{M}(\Vert u\Vert }_{L^2(\Omega )}^2)\mathrm{strongly}\mathrm{in}{L}^2(0,T)\hfill \\ u_{m}ln\mid u_m{\mid }^{\theta }⟶uln\mid u{\mid }^{\theta }\mathrm{a}.\mathrm{e}.\mathrm{in}\mathcal{Q}\hfill \end{array}$$

(47)

Finally, from Equation (35) we have (using the last result of Equation (47) and dominated convergence theorem)

$$u_{m}ln\mid u_m{\mid }^{\theta }\rightharpoonup uln\mid u{\mid }^{\theta }\mathrm{weakly}\mathrm{in}{L}^2(\mathcal{Q}).$$

(48)

By using Equations (45)–(48) we can pass to the limit in a standard way in Equation (20), where the limit solution $(u;w^i)$ satisfies the system (for all $(v,{\psi }^i)\in (H_0^1(\Omega )\cap H^2(\Omega ))\times L^2(\Omega \times ({\alpha }_i,{\beta }_i)\times (0,1))$)

$$\begin{array}{c}{\displaystyle \left(\frac{{\partial }^{2}u}{\partial t^2},v\right)-{\mathcal{K}(\Vert \sigma \nabla u\Vert }_{L^2(\Omega )}^2)\left(div(A_{\sigma }\nabla u,v\right)+\left(\mathcal{M}(\Vert u{\Vert }_{L^2(\Omega )}^2)u,v\right)}\hfill \\ {\displaystyle -\zeta (t)\left(div(A_{\sigma }\nabla \frac{\partial u}{\partial t}),v\right)+d_0(t)\left(\frac{\partial u}{\partial t},v\right)}\hfill \\ {\displaystyle +\sum _{j=1}^n\left(d_j(t){\int }_{{\alpha }_j}^{{\beta }_j}{e}_i^2(\mathbf{x},\tau )w^j(\mathbf{x},t,\tau ;1)d\tau ,v\right)=\left(uln\mid u{\mid }^{\theta },v\right),}\hfill \\ {\displaystyle {\int }_0^1\left(c_i(t,\tau )\frac{\partial w^i}{\partial t}(.,t,\tau ;\eta ),{\psi }^i(.,\tau ;\eta )\right)d\eta }\hfill \\ {\displaystyle +{\int }_0^1\left(\left(1-\frac{\partial c_i}{\partial t}(t,\tau )\right)\frac{\partial w^i}{\partial \eta }(.,t,\tau ;\eta ),{\psi }^i(.,\tau ;\eta )\right)d\eta =0,}\hfill \\ {\displaystyle u(\mathbf{x},0)={\varphi }_0(\mathbf{x}),\frac{\partial u}{\partial t}(\mathbf{x},0)={\varphi }_1(\mathbf{x}),\mathrm{for}\mathbf{x}\in \Omega }\hfill \\ {\displaystyle w^i(\mathbf{x},0,\tau ;\eta )=w_0^i(\mathbf{x},\tau ,\eta ),\mathrm{in}\Omega \times (0,{\beta }_i)\times (0,1),}\hfill \\ {\displaystyle w^i(\mathbf{x},t,\tau ;0)=\frac{\partial u}{\partial t}(\mathbf{x},t),\mathrm{in}\mathcal{Q}\times ({\alpha }_i,{\beta }_i),}\hfill \\ {\displaystyle u(\mathbf{x},t)=0,for(\mathbf{x},t)\in \Sigma .}\hfill \end{array}$$

(49)

The proof of the local existence of the solution is now completed. Prove now the second part of the theorem.

In the same way as to obtain Equation (28), we obviously get

$$\begin{array}{c}{\displaystyle \frac{1}{2}\frac{d\mathcal{E}}{dt}(t)+\zeta (t)\Vert \sigma \nabla \frac{\partial u}{\partial t}{\Vert }_{L^2(\Omega )}^2\le -{\aleph }_1(\lambda ){\Vert \frac{\partial u}{\partial t}\Vert }_{L^2(\Omega )}^2}\hfill \\ {\displaystyle -{\aleph }_2(\lambda )\sum _{j=1}^n{\int }_{\Omega }{\int }_{{\alpha }_j}^{{\beta }_j}{e}_j^2(\mathbf{x},\tau ){(w^j)}^2(\mathbf{x},t,\tau ;1)d\mathbf{x}d\tau }\hfill \\ {\displaystyle \le 0,}\hfill \end{array}$$

(50)

where ${\aleph }_1(\lambda )$ and ${\aleph }_2(\lambda )$ are given in Equation (18), and ${\displaystyle w^i(\mathbf{x},t,\tau ;1)=\frac{\partial u}{\partial t}(\mathbf{x},t-c_i(t,\tau ))}$ (from Equation (13)). Then, the function $\mathcal{E}$ is nonincreasing. □

We now present some examples illustrating the conditions for the existence of a solution.

Example 1.

For the weak damping coefficient $d_0$ we can take, for example, $d_0(t)=b_1+b_{2}cos(\omega t)$, with $b_1> \mid b_2\mid$ and $\omega >0$ or $d_0(t)=b_1+{\displaystyle \frac{b_2}{1+\omega t}}$, with $b_1> \mid min(0,b_2)\mid$ and $\omega >0$ or even $d_0(t)=b_1+b_2{e}^{-\omega {(t-{\tau }_0)}^2}$, with $b_1>\mid min(0,b_2)\mid$ and $\omega >0$. We denote by $d^{inf}$ the following value: ${\displaystyle 0<d^{inf}=\underset{s\ge 0}{inf}{d}_0(s)}$. Concerning the strong damping coefficient ζ, we can consider one of the following functions:

• $\zeta (t)=a_1{e}^{-{\omega }_{1}t}+(1-a_1)e^{-{\omega }_2{t}^2}$, with $a_1\in ]0,1]$ and ${\omega }_i>0$, for $i=1,2$.
  In this case, we have ${\xi }^{\prime }(t)=-a_1{\omega }_1{e}^{-{\omega }_{1}t}-2(1-a_1){\omega }_{2}t{e}^{-{\omega }_2{t}^2}<0$.
• $\zeta (t)=a_1-a_2{e}^{-\omega t}$ and ${\zeta }_0\in [0,1[$, with $a_1>a_2\ge 0$, $\omega >0$ and $0\le {\displaystyle \frac{a_2\omega }{2{\vartheta }_1}}\le {\zeta }_0<1$.
  In this case, we have ${\xi }^{\prime }(t)=a_2\omega e^{-{\omega }_{1}t}\le 2{\vartheta }_1{\xi }_0$.
• $\zeta (t)=a_1-a_{2}cos\omega t$ and ${\zeta }_0\in [0,1[$, with $a_1>a_2\ge 0$, $\omega >0$ and $0\le {\displaystyle \frac{a_2\omega }{2{\vartheta }_1}}\le {\zeta }_0<1$.
  In this case, we have ${\xi }^{\prime }(t)=a_2\omega sin\omega t<2{\vartheta }_1{\zeta }_0$.

For the time-varying delays functions $c_i$, for $i=1, n$, we can take these functions, for example, in the form $c_i(t,\tau )=a_i-b_{i}arctan({\omega }_i(t+ϵ))e^{-{\gamma }_i\tau }$, with $a_i>0$, $b_i>0$, ${\omega }_i>0$, ${\gamma }_i>0$, for $i=1, n$, and $ϵ>0$ a small parameter. The involved parameters $a_i$, $b_i$ and ${\gamma }_i$ are chosen appropriately to guarantee the positivity of $c_i$. In this case, we have $1-{\displaystyle \frac{\partial c_i}{\partial t}}=1+{\displaystyle \frac{b_i{\omega }_i}{1+{({\omega }_i(t+ϵ))}^2}}{e}^{-{\gamma }_i\tau }>0$ and ${\displaystyle \frac{\partial c_i}{\partial \tau }}=b_i{\gamma }_{i}arctan({\omega }_i(t+ϵ))e^{-{\gamma }_i\tau }>0$. Then, we get

$${\displaystyle {(1-\frac{\partial c_i}{\partial t})}^{-1}=\frac{1+{({\omega }_i(t+ϵ))}^2}{1+{({\omega }_i(t+ϵ))}^2+b_i{\omega }_i{e}^{-\tau }}<1.}$$

(51)

For the bounded functions $d_i$ and $e_i$, for $i=1,n$, we denote by $E^{sup}$ and $D^{sup}$ the values

$${\displaystyle E^{sup}=\sum _{i=1}^n{\int }_{{\alpha }_i}^{{\beta }_i}{\Vert e_i^2(.,\tau )\Vert }_{L^{\infty }(\Omega )}d\tau \mathrm{and}{D}^{sup}=\underset{i=1,n}{max}(\underset{s\ge 0}{sup}\mid d_i(s)\mid )}$$

(52)

and we assume that

$$d^{inf}>D^{sup}{E}^{sup}.$$

(53)

We can now estimate the values ${{\rm Y}}_1$ and ${{\rm Y}}_2$.

From Equation (52), we have (for all $\mathbf{x}\in \Omega$, $s\ge 0$ and $i\in \left[\right[1,n\left]\right]$)

$${\displaystyle \sum _{j=1}^n{\int }_{{\alpha }_i}^{{\beta }_i}{e}_j^2(\mathbf{x},\tau )d\tau \le E^{sup}\mathrm{and}\mid d_i(s)\mid \le D^{sup}}$$

(54)

and then (according to Equations (51) and (54))

$$\begin{array}{c}{\displaystyle \frac{{\displaystyle 2d_0(t)-\sum _{j=1}^n\mid d_j(t)\mid {\int }_{{\alpha }_j}^{{\beta }_j}{e}_j^2(\mathbf{x},\tau )d\tau }}{{\displaystyle \sum _{j=1}^n{\int }_{{\alpha }_i}^{{\beta }_i}{e}_j^2(\mathbf{x},\tau )d\tau }}\ge \frac{{\displaystyle 2d^{inf}-D^{sup}{E}^{sup}}}{{\displaystyle E^{sup}}},}\hfill \\ {\displaystyle \mid d_j(t)\mid {(1-\frac{\partial c_j}{\partial t}(t,\tau ))}^{-1}\le D^{sup}.}\hfill \end{array}$$

(55)

Consequently,

$$\begin{array}{c}{{\rm Y}}_1\ge {\displaystyle \frac{{\displaystyle 2d^{inf}-D^{sup}{E}^{sup}}}{{\displaystyle E^{sup}}}},\hfill \\ {{\rm Y}}_2\le D^{sup}.\hfill \end{array}$$

(56)

From Equation (53), we can deduce the following estimates:

$$\begin{array}{c}{{\rm Y}}_1\ge {\displaystyle \frac{{\displaystyle 2d^{inf}-D^{sup}{E}^{sup}}}{{\displaystyle E^{sup}}}}>D^{sup}\ge {{\rm Y}}_2\hfill \end{array}$$

and then

$$\begin{array}{c}{{\rm Y}}_1>{{\rm Y}}_2.\hfill \end{array}$$

(57)

Now, we show the global existence of solution and exponential decay of the energy for Equation (14).

3. Potential Well and Stable Set

This section deals with the potential well and the construction of a stable set for the problem in order to provide a global solution.

We start by introducing the useful functionals $\mathcal{I}$ and $\mathcal{F}$ such that, for $v\in H_0^1(\Omega )/\left\{0\right\}$

$$\begin{array}{c}{\displaystyle \mathcal{I}(v)={\vartheta }_1{\Vert \sigma \nabla v\Vert }_{L^2(\Omega )}^2+\frac{{\vartheta }_0}{2}{\Vert v\Vert }_{L^2(\Omega )}^2-\theta {\int }_{\Omega }{v}^{2}ln\mid v\mid d\mathbf{x}}\hfill \\ {\displaystyle \mathcal{F}(v)=\mathcal{I}(v)+\frac{\theta }{2}{\Vert v\Vert }_{L^2(\Omega )}^2.}\hfill \end{array}$$

For any $r>0$ and $v\in H_0^1(\Omega )/\left\{0\right\}$, we have

$$\mathcal{F}(rv)=\mathcal{I}(rv)+{\displaystyle \frac{r^2\theta }{2}}{\Vert v\Vert }_{L^2(\Omega )}^2=r^2(\mathcal{I}(v)+{\displaystyle \frac{\theta (1-2ln(r))}{2}}{\Vert v\Vert }_{L^2(\Omega )}^2).$$

Lemma 3.

Let v be given in $H_0^1(\Omega )/\left\{0\right\}$, we have the following results:

(i) 

For any $r>0$, ${\displaystyle r\frac{\partial }{\partial r}(\mathcal{F}(rv))=2\mathcal{I}(rv)}$.

(ii) 

${\displaystyle arg\underset{r>0}{max}\mathcal{F}(rv)={\overline{r}}_{\theta }(v)}$ and $\mathcal{I}({\overline{r}}_{\theta }(v)v)=0$, with ${\displaystyle {\overline{r}}_{\theta }(v)=exp(\frac{\mathcal{I}(v)}{{\theta \Vert v\Vert }_{L^2(\Omega )}^2})}$.

Proof. 

(i) A simple calculation shows that

$$\begin{array}{c}{\displaystyle \frac{\partial }{\partial r}(\mathcal{F}(rv))=2r(\mathcal{I}(v)+\frac{\theta (1-2ln(r))}{2}{\Vert v\Vert }_{L^2(\Omega )}^2{)-r\theta \Vert v\Vert }_{L^2(\Omega )}^2,}\hfill \\ {\displaystyle =2r(\mathcal{I}(v)-\theta ln(r)\Vert v{\Vert }_{L^2(\Omega )}^2)}\hfill \end{array}$$

and then ${\displaystyle r\frac{\partial }{\partial r}(\mathcal{F}(rv))=2\mathcal{I}(rv)}$. (ii) Since ${\displaystyle \underset{r\to 0^{+}}{lim}\mathcal{F}(rv)=0}$, ${\displaystyle \underset{r\to +\infty }{lim}\mathcal{F}(rv)=-\infty }$, and ${\displaystyle \frac{\partial }{\partial r}\mathcal{F}(rv)>0}$ if $0<r<{\overline{r}}_{\theta }(v)$, ${\displaystyle \frac{\partial }{\partial r}\mathcal{F}(rv)<0}$ if $r>{\overline{r}}_{\theta }(v)$ and ${\displaystyle \frac{{\partial }^2}{\partial r^2}{\mathcal{F}(rv)|}_{r={\overline{r}}_{\theta }(v)}=-2\theta {\Vert v\Vert }_{L^2(\Omega )}^2<0}$, where ${\displaystyle {\overline{r}}_{\theta }(v)=exp(\frac{\mathcal{I}(v)}{{\theta \Vert v\Vert }_{L^2(\Omega )}^2})}$ is the unique solution of ${\displaystyle \frac{\partial }{\partial r}\mathcal{F}(rv)=0}$, then ${\displaystyle arg\underset{r>0}{max}\mathcal{F}(rv)={\overline{r}}_{\theta }(v)}$ and $\mathcal{I}({\overline{r}}_{\theta }(v)v)=0$. □

Associated with the function $\mathcal{F}$, we can define the well-known Nehari Manifold by

$${\displaystyle \mathcal{N}=\{v\in {\mathbb{V}}_0(\Omega )/\left\{0\right\}:(\frac{\partial }{\partial r}\mathcal{F}(rv)){|}_{r=1}=0\},}$$

equivalently,

$${\displaystyle \mathcal{N}=\{v\in {\mathbb{V}}_0(\Omega )/\left\{0\right\}:\mathcal{I}(v)=0\},}$$

where ${\mathbb{V}}_0(\Omega )=H_0^1(\Omega )\cap H^2(\Omega )$.

As in [56], the potential well depth d (also known as mountain pass level), is defined as

$${\displaystyle d=\underset{v\in {\mathbb{V}}_0(\Omega )/\left\{0\right\}}{inf}\underset{r>0}{sup}\mathcal{F}(rv).}$$

(58)

This value d may also be described as (as noted in [57])

$${\displaystyle d=\underset{v\in \mathcal{N}}{inf}\mathcal{F}(v).}$$

(59)

Lemma 4.

The value of the potential well depth d satisfies the following relations:

$$\begin{array}{c}{\displaystyle d\ge d_I=\frac{\theta }{2}exp\left(\frac{2{\mu }_{\theta }(\alpha ){\tilde{C}}_{\Omega ,\sigma }^2+{\vartheta }_0+N\theta (1+ln\alpha )}{\theta }\right)}\hfill \\ {\displaystyle =\frac{\theta }{2}exp\left(\frac{2{\mu }_{\theta }(\alpha ){\tilde{C}}_{\Omega ,\sigma }^2+{\vartheta }_0}{\theta }\right)e^N{\alpha }^N,}\hfill \\ {\displaystyle d_I\ge d_A=\frac{\theta }{2}exp\left(\frac{{\vartheta }_0}{\theta }\right)e^N{\alpha }^N,}\hfill \end{array}$$

(60)

where ${\mu }_{\theta }(\alpha )={\vartheta }_1-{\displaystyle \frac{\theta {\alpha }^2}{2\pi {\underline{\varsigma }}^2}}$ and ${\tilde{C}}_{\Omega ,\sigma }=C_{\Omega }^{-1}\underline{\varsigma }$ (with $C_{\Omega }$ the Poincaré constant).

Furthermore, we have

$$\begin{array}{c}{\displaystyle {\rho }_0=\frac{\theta }{2}exp(\frac{2{\mu }_{\theta }({\alpha }_1){\tilde{C}}_{\Omega ,\sigma }^2+{\vartheta }_0}{\theta })e^N{\alpha }_0^N<d_I<{\rho }_1=\frac{\theta }{2}exp(\frac{2{\mu }_{\theta }({\alpha }_0){\tilde{C}}_{\Omega ,\sigma }^2+{\vartheta }_0}{\theta })e^N{\alpha }_1^N,}\hfill \\ {\displaystyle {\rho }_{0a}=\frac{\theta }{2}exp(\frac{{\vartheta }_0}{\theta })e^N{\alpha }_0^N<d_A<{\rho }_{1a}=\frac{\theta }{2}exp(\frac{{\vartheta }_0}{\theta })e^N{\alpha }_1^N,}\hfill \end{array}$$

(61)

where ${\displaystyle {\mu }_{\theta }({\alpha }_1)={\vartheta }_1-\frac{\theta {\alpha }_1^2}{2\pi {\underline{\varsigma }}^2}}$ and ${\displaystyle {\mu }_{\theta }({\alpha }_0)={\vartheta }_1-\frac{\theta {\alpha }_0^2}{2\pi {\underline{\varsigma }}^2}}$.

Proof. 

From assumption (H5), Lemmas 1 and 3 and Equation (7), we obtain (for all $v\in {\mathbb{V}}_0(\Omega )/\left\{0\right\}$)

$$\begin{array}{c}{\displaystyle 0=\mathcal{I}({\overline{r}}_{\theta }(v)v)}\hfill \\ {\displaystyle \ge \left({\mu }_{\theta }(\alpha ){\tilde{C}}_{\Omega ,\sigma }^2+\frac{N\theta }{2}(1+ln\alpha )+\frac{{\vartheta }_0}{2}-\frac{\theta }{2}ln{\Vert {\overline{r}}_{\theta }(v)v\Vert }_{L^2(\Omega )}^2\right){\Vert {\overline{r}}_{\theta }(v)v\Vert }_{L^2(\Omega )}^2}\hfill \end{array}$$

and then

$${\displaystyle \Vert {\overline{r}}_{\theta }{(v)v\Vert }_{L^2(\Omega )}^2\ge exp(\frac{2{\mu }_{\theta }(\alpha ){\tilde{C}}_{\Omega ,\sigma }^2+N\theta (1+ln\alpha )+{\vartheta }_0}{\theta }).}$$

This implies

$$\begin{array}{c}{\displaystyle d\ge \mathcal{F}({\overline{r}}_{\theta }(v)v)=\frac{\theta }{2}{\Vert {\overline{r}}_{\theta }(v)v\Vert }_{L^2(\Omega )}^2\ge d_I=\frac{\theta }{2}exp(\frac{2{\mu }_{\theta }(\alpha ){\tilde{C}}_{\Omega ,\sigma }^2+N\theta (1+ln\alpha )+{\vartheta }_0}{\theta })}\hfill \\ {\displaystyle \ge d_A=\frac{\theta }{2}exp(\frac{{\vartheta }_0}{\theta })e^N{\alpha }^N.}\hfill \end{array}$$

From Equation (10), we can easily deduce Equation (61). □

Now, we can define the stable set ${\mathcal{W}}_s$ and unstable set ${\mathcal{W}}_{un}$, respectively by (see [57,58] and also, e.g., [59])

$$\begin{array}{c}{\displaystyle {\mathcal{W}}_s=\{v\in {\mathcal{N}}^{+}:\mathcal{F}(v)<d\},}\hfill \\ {\displaystyle {\mathcal{W}}_{un}=\{v\in {\mathcal{N}}^{-}:\mathcal{F}(v)<d\},}\hfill \end{array}$$

(62)

where

$$\begin{array}{c}{\displaystyle {\mathcal{N}}^{+}=\{v\in {\mathbb{V}}_0(\Omega )/\left\{0\right\}:\mathcal{I}(v)>0\}\cup \left\{0\right\},}\hfill \\ {\displaystyle {\mathcal{N}}^{-}=\{v\in {\mathbb{V}}_0(\Omega )/\left\{0\right\}:\mathcal{I}(v)<0\},}\hfill \end{array}$$

Lemma 5.

Let v be in $H_0^1(\Omega )\setminus \left\{0\right\}$ and ${\chi }_{min}$ be the first eigenvalue of $-div(\sigma \nabla .)$ subject to homogeneous Dirichlet boundary conditions, which may also be defined by

$${\displaystyle {\chi }_{min}=\underset{v\in H_0^1(\Omega )\setminus \left\{0\right\}}{inf}\frac{{\Vert \sigma \nabla v\Vert }_{L^2(\Omega )}^2}{{\Vert v\Vert }_{L^2(\Omega )}^2}.}$$

Then, we have the following:

(i) 

If ${\displaystyle {0<\Vert \sigma \nabla v\Vert }_{L^2(\Omega )}^2\le {\chi }_{min}{S}_N(\frac{{\vartheta }_0}{\theta },\frac{{\vartheta }_1{\underline{\varsigma }}^2}{\theta })}$ then $\mathcal{I}(v)\ge 0$,

(ii) 

If ${\displaystyle \mathcal{I}(v)<0}$, then ${\displaystyle {\Vert \sigma \nabla v\Vert }_{L^2(\Omega )}^2>{\chi }_{min}{S}_N(\frac{{\vartheta }_0}{\theta },\frac{{\vartheta }_1{\underline{\varsigma }}^2}{\theta })}$,

where ${\displaystyle S_N(\frac{{\vartheta }_0}{\theta },\frac{{\vartheta }_1{\underline{\varsigma }}^2}{\theta })=e^{N+\frac{{\vartheta }_0}{\theta }}{\left(2\pi \frac{{\vartheta }_1{\underline{\varsigma }}^2}{\theta }\right)}^{\frac{N}{2}}}$.

Proof. 

It follows from Lemma 1, for any constant $\alpha >0$, that ($\forall v\in H_0^1(\Omega )\setminus \left\{0\right\}$)

$$\begin{array}{c}{\displaystyle \mathcal{I}(v)={\vartheta }_1{\Vert \sigma \nabla v\Vert }_{L^2(\Omega )}^2+\frac{{\vartheta }_0}{2}{\Vert v\Vert }_{L^2(\Omega )}^2-\theta {\int }_{\Omega }{v}^{2}ln\mid v\mid dx}\hfill \\ {\displaystyle \ge \left({\vartheta }_1-\frac{\theta {\alpha }^2}{2\pi {\underline{\varsigma }}^2}\right){\Vert \sigma \nabla v\Vert }_{L^2(\Omega )}^2}\hfill \\ {\displaystyle +\frac{\theta }{2}{\Vert v\Vert }_{L^2(\Omega )}^2\left(\left(N(1+ln\alpha )+\frac{{\vartheta }_0}{\theta }\right)-ln{\Vert v\Vert }_{L^2(\Omega )}^2\right).}\hfill \end{array}$$

For $\alpha$ such that ${\displaystyle {\vartheta }_1-\frac{\theta {\alpha }^2}{2\pi {\underline{\varsigma }}^2}=ϵ{\vartheta }_1}$, with $0<ϵ<1$ (i.e., ${\displaystyle \alpha =\underline{\varsigma }\sqrt{\frac{2\pi {\vartheta }_1^{ϵ}}{\theta }}}$, with ${\displaystyle {\vartheta }_1^{ϵ}={\vartheta }_1(1-ϵ)}$), we can deduce that (since ${\displaystyle {\vartheta }_1^{ϵ}={\vartheta }_1(1-ϵ)\to {\vartheta }_1}$ when $ϵ$ tends towards 0)

$$\begin{array}{c}{\displaystyle \mathcal{I}(v)\ge \frac{\theta }{2}{\Vert v\Vert }_{L^2(\Omega )}^2\left(\frac{N}{2}ln(\frac{2\pi {\vartheta }_1{\underline{\varsigma }}^2}{\theta })+(N+\frac{{\vartheta }_0}{\theta })-ln{\Vert v\Vert }_{L^2(\Omega )}^2\right).}\hfill \end{array}$$

From this, we derive easily the results (i) and (ii). □

Lemma 6.

Suppose that the assumptions of Theorem 1 hold and let $({\varphi }_0,{\varphi }_1,h_0)$ be in ${\mathcal{H}}_0({\mathbb{S}}_0)$. Then, for $E_a\in ]0,d]$, the following results hold:

(i) 

If there exists $t_1\in [0,T_{max})$ such that $u(t_1)\in {\mathcal{W}}_{un}$ and $\mathcal{E}(t_1)<E_a\le d$, then $u(t)\in {\mathcal{W}}_{un}$, $\forall t\in [t_1,T_{max})$. Moreover, we have ($\forall t\in [t_1,T_{max})$)

$${\displaystyle \frac{\theta }{2}}{\Vert u(t)\Vert }_{L^2(\Omega )}^2>E_a.$$

(63)

(ii) 

If there exists $t_1\in [0,T_{max})$ such that $u(t_1)\in {\mathcal{W}}_s$ and $\mathcal{E}(t_1)<E_a\le d$, then $u(t)\in {\mathcal{W}}_s$, $\forall t\in [t_1,T_{max}).$ Moreover, we have ($\forall t\in [t_1,T_{max})$)

$${\displaystyle \frac{\theta }{2}}{\Vert u(t)\Vert }_{L^2(\Omega )}^2<E_a,$$

(64)

$T_{max}$ is the maximum existence time of weak solution to Equation (1).

Proof. 

By using Equation (15) of the energy $\mathcal{E}$ and a similar argument to derive Lemma A3 in [24], we can derive (i) and (ii) of the lemma. So we omit the details. □

4. Global Solution

In this section we prove that the local weak solution to Equation (14) is a global solution for subcritical level of initial energy.

Theorem 2.

Suppose that the hypotheses of Theorem 1 hold and that $({\varphi }_0,{\varphi }_1,h_0)\in {\mathcal{H}}_0({\mathbb{S}}_0)$. Let $E_a\in ]0,d]$. If ${\varphi }_0\in {\mathcal{W}}_s$ and $\mathcal{E}(0)<E_a$, then the local weak solution u to Equation (14) is a global solution.

Proof. 

First, according to Lemma 6, we get that $u(t)\in {\mathcal{W}}_s$, $\forall t\in [0,T_{max})$. Show now that $T_{max}=\infty$. For this, it is sufficient to demonstrate the boundedness of ${\displaystyle \Vert \frac{\partial u}{\partial t}{\Vert }_{L^2(\Omega )}^2}$ and of ${\Vert \sigma \nabla u\Vert }_{L^2(\Omega )}^2$.

Now, since that $0<\mathcal{E}(0)<E_a\le d$, we obtain $\forall t\in [0,T_{max})$ (according to the positivity of $\mathcal{I}(u(t))$ and expression of $\mathcal{E}(t)$)

$$\begin{array}{c}{\displaystyle \Vert \frac{\partial u}{\partial t}{\Vert }_{L^2(\Omega )}^2\le \mathcal{E}(t)\le \mathcal{E}(0)<E_a,}\hfill \\ {\displaystyle \frac{\theta }{2}{\Vert u\Vert }_{L^2(\Omega )}^2\le \mathcal{E}(t)\le \mathcal{E}(0)<E_a.}\hfill \end{array}$$

(65)

It follows from Lemma 1 that

$$\begin{array}{c}{\displaystyle {\vartheta }_1{\Vert \sigma \nabla u\Vert }_{L^2(\Omega )}^2=\mathcal{I}(u(t))-\frac{{\vartheta }_0}{2}{\Vert u\Vert }_{L^2(\Omega )}^2+\theta {\int }_{\Omega }{u}^{2}ln\mid u\mid d\mathbf{x}}\hfill \\ {\displaystyle \le \mathcal{I}(u(t))+\frac{\theta }{2}{\Vert u\Vert }_{L^2(\Omega )}^2\left({ln(\Vert u\Vert }_{L^2(\Omega )}^2)-N(1+ln\alpha )-\frac{{\vartheta }_0}{\theta }\right)}\hfill \\ {\displaystyle +\frac{\theta {\alpha }^2}{2\pi {\underline{\varsigma }}^2}{\Vert \sigma \nabla u\Vert }_{L^2(\Omega )}^2}\hfill \\ {\displaystyle \le \mathcal{E}(t)+\frac{\theta }{2}{\Vert u\Vert }_{L^2(\Omega )}^2\left({ln(\Vert u\Vert }_{L^2(\Omega )}^2)-N(1+ln\alpha )-\frac{{\vartheta }_0}{\theta }\right)}\hfill \\ {\displaystyle +\frac{\theta {\alpha }^2}{2\pi {\underline{\varsigma }}^2}{\Vert \sigma \nabla u\Vert }_{L^2(\Omega )}^2}\hfill \end{array}$$

(66)

and then

$$\begin{array}{c}{\displaystyle {\mu }_{\theta }{(\alpha )\Vert \sigma \nabla u\Vert }_{L^2(\Omega )}^2\le \mathcal{E}(t)+\frac{\theta }{2}{\Vert u\Vert }_{L^2(\Omega )}^2{g}_{\frac{{\vartheta }_0}{\theta },\alpha }{(\Vert u\Vert }_{L^2(\Omega )}),}\hfill \end{array}$$

(67)

where ${\mu }_{\theta }(\alpha )={\vartheta }_1-{\displaystyle \frac{\theta {\alpha }^2}{2\pi {\underline{\varsigma }}^2}}>0$ and

$$g_{{\displaystyle \frac{{\vartheta }_0}{\theta }},\alpha }{(\Vert u\Vert }_{L^2(\Omega )}{)=ln(\Vert u\Vert }_{L^2(\Omega )}^2)-N(1+ln\alpha )-{\displaystyle \frac{{\vartheta }_0}{\theta }}.$$

(68)

From Equations (60) (in Lemma 4) and (65) and (H5), we can get that

$$\begin{array}{c}{\displaystyle g_{\frac{{\vartheta }_0}{\theta },\alpha }{(\Vert u\Vert }_{L^2(\Omega )})\le ln(\frac{2d}{\theta })-N(1+ln\alpha )-\frac{{\vartheta }_0}{\theta },}\hfill \\ {\displaystyle ln(\frac{2d}{\theta })-N(1+ln\alpha )-\frac{{\vartheta }_0}{\theta }\ge ln(\frac{2d_I}{\theta })-N(1+ln\alpha )-\frac{{\vartheta }_0}{\theta }}\hfill \\ {\displaystyle \ge \frac{2{\mu }_{\theta }(\alpha ){\tilde{C}}_{\Omega ,\sigma }^2}{\theta }+N(1+ln\alpha )-N(1+ln\alpha )-\frac{{\vartheta }_0}{\theta }+\frac{{\vartheta }_0}{\theta }}\hfill \\ {\displaystyle =\frac{2{\mu }_{\theta }(\alpha ){\tilde{C}}_{\Omega ,\sigma }^2}{\theta }>0.}\hfill \end{array}$$

(69)

Hence (because $\mathcal{E}(t)\le \mathcal{E}(0)<d)$

$$\begin{array}{c}{\displaystyle {\Vert \sigma \nabla u\Vert }_{L^2(\Omega )}^2<d\left[1+ln(2d/\theta )-N(1+ln\alpha )-\frac{{\vartheta }_0}{\theta }\right]{\mu }_{\theta }{(\alpha )}^{-1}.}\hfill \end{array}$$

(70)

Therefore, from Equations (65) and (70), we can conclude that u is a global solution (by the continue principle). □

Remark 2.

(a) From Equation (50), we can obtain the inequalities (since $0<\mathcal{E}(t)\le \mathcal{E}(0)<d$, for $t\ge 0$)

$$\begin{array}{c}{\displaystyle {\int }_0^t{\Vert \frac{\partial u}{\partial t}\Vert }_{L^2(\Omega )}^{2}ds\le \frac{d}{2{\aleph }_1(\lambda )},}\hfill \\ {\displaystyle \sum _{i=1}^n{\int }_0^t\left({\int }_{\Omega }{\int }_{{\alpha }_i}^{{\beta }_i}{e}_i^2(\mathbf{x},\tau ){(w^i)}^2(\mathbf{x},t,\tau ;1)d\mathbf{x}d\tau \right)ds\le \frac{d}{2{\aleph }_2(\lambda )}.}\hfill \end{array}$$

(71)

(b) 

The results of Theorem 2 still remain valid when the conditions ${\varphi }_0\in {\mathcal{W}}_s$ and $\mathcal{E}(0)<E_a$ are replaced by the existence of a real number $t_1\in [0;T_{max})$ with $u(t_1)\in {\mathcal{W}}_s$ and $\mathcal{E}(t_1)<E_a$.

(c) 

From Equations (60) (in Lemma 4) and (65) and (H5), the following inequalities hold:

(i) 

If $E_a\ge d_I$

$$\begin{array}{c}{\displaystyle g_{\frac{{\vartheta }_0}{\theta },\alpha }{(\Vert u\Vert }_{L^2(\Omega )})\le ln(\frac{2E_a}{\theta })-N(1+ln\alpha )-\frac{{\vartheta }_0}{\theta }\mathrm{and}}\hfill \\ {\displaystyle ln(\frac{2E_a}{\theta })-N(1+ln\alpha )-\frac{{\vartheta }_0}{\theta }\ge ln(\frac{2d_I}{\theta })-N(1+ln\alpha )-\frac{{\vartheta }_0}{\theta }}\hfill \\ {\displaystyle =\frac{2{\mu }_{\theta }(\alpha ){\tilde{C}}_{\Omega ,\sigma }^2}{\theta }>0.}\hfill \end{array}$$

(72)

(ii) 

If $E_a\le d_I$

$$\begin{array}{c}{\displaystyle g_{\frac{{\vartheta }_0}{\theta },\alpha }{(\Vert u\Vert }_{L^2(\Omega )})\le ln(\frac{2E_a}{\theta })-N(1+ln\alpha )-\frac{{\vartheta }_0}{\theta }}\hfill \\ {\displaystyle \le ln(\frac{2d_I}{\theta })-N(1+ln\alpha )-\frac{{\vartheta }_0}{\theta }=\frac{2{\mu }_{\theta }(\alpha ){\tilde{C}}_{\Omega ,\sigma }^2}{\theta },}\hfill \end{array}$$

(73)

(iii) 

if $E_a\le d_A$

$$\begin{array}{c}{\displaystyle g_{\frac{{\vartheta }_0}{\theta },\alpha }{(\Vert u\Vert }_{L^2(\Omega )})\le ln(\frac{2E_a}{\theta })-N(1+ln\alpha )-\frac{{\vartheta }_0}{\theta }}\hfill \\ {\displaystyle \le ln(\frac{2d_A}{\theta })-N(1+ln\alpha )-\frac{{\vartheta }_0}{\theta }=0,}\hfill \end{array}$$

(74)

where ${\displaystyle {\mu }_{\theta }(\alpha )={\vartheta }_1-\frac{\theta {\alpha }^2}{2\pi {\underline{\varsigma }}^2}}$ and ${\tilde{C}}_{\Omega ,\sigma }=C_{\Omega }^{-1}\underline{\varsigma }$ with $C_{\Omega }$ the Poincaré constant.

5. Exponential Decay

In this section, we establish the exponential decay property of Equation (14) by appropriate integral inequalities (according to a similar principle as in [60]). For this, we impose that the operators $\mathcal{M}$ and $\mathcal{K}$ satisfy the following conditions ($\forall y\ge 0$)

$$\begin{array}{c}{\displaystyle \mathcal{M}(y)y-{\mathcal{M}}_p(y)\ge 0,\mathcal{K}(y)y-{\mathcal{K}}_p(y)\ge 0.}\hfill \end{array}$$

(75)

Theorem 3.

Let assumptions of Theorem 2 hold and assume that ${\varphi }_0\in {\mathcal{W}}_s$ and $\mathcal{E}(0)<E_a$ with $E_a\le d_I$. Then, if $E_a\le d_A$ or if $E_a\ge d_A$ with (H_a) ${\displaystyle M(v)\ge \frac{(1+{\varpi }_a){\vartheta }_0}{2}>\frac{{\vartheta }_0}{2},\forall v\ge 0,}$ and ${\displaystyle 2{\mu }_{\theta }{\tilde{C}}_{\Omega ,\sigma }^2\le {\varpi }_a{\vartheta }_0}$, for some ${\varpi }_a>0$, there exist constants $C_{kc}>0$ and $\delta >0$ such that the energy $\mathcal{E}$ satisfies the relation

$$0<\mathcal{E}(t)\le C_{kc}{e}^{-\delta t},\forall t\ge 0.$$

Proof. 

Taking $v=u$ and ${\displaystyle {\psi }^i=exp(-\eta c_i(t,\tau ))e_i^2(\mathbf{x},\tau )w^i}$ in Equation (49), we get that

$$\begin{array}{c}{\displaystyle -\Vert \frac{\partial u}{\partial t}{\Vert }_{L^2(\Omega )}^2+\frac{d}{dt}(\frac{\partial u}{\partial t},u)+{\mathcal{K}(\Vert \sigma \nabla u\Vert }_{L^2(\Omega )}^2{)\Vert \sigma \nabla u\Vert }_{L^2(\Omega )}^2}\hfill \\ {\displaystyle +{\mathcal{M}(\Vert u\Vert }_{L^2(\Omega )}^2{)\Vert u\Vert }_{L^2(\Omega )}^2+\zeta (t)(\sigma \nabla \frac{\partial u}{\partial t},\sigma \nabla u)+d_0(t)(\frac{\partial u}{\partial t},u)}\hfill \\ {\displaystyle +\sum _{j=1}^n{\int }_{\Omega }{d}_j(t){\int }_{{\alpha }_j}^{{\beta }_j}{e}_i^2(\mathbf{x},\tau )w^j(\mathbf{x},t,\tau ;1)u(\mathbf{x},t)d\tau d\mathbf{x}={\int }_{\Omega }{u}^{2}ln\mid u{\mid }^{\theta }d\mathbf{x}}\hfill \end{array}$$

(76)

and by applying a simple manipulation of the derivatives in the second part of Equation (49) (since $w^i(.;\eta =0)={\displaystyle \frac{\partial u}{\partial t}}$)

$$\begin{array}{c}{\displaystyle {\int }_0^1{\int }_{\Omega }{\int }_{{\alpha }_i}^{{\beta }_i}\frac{\partial }{\partial t}\left[c_i(t,\tau )exp(-\eta c_i(t,\tau ))e_i^2(\mathbf{x},\tau ){(w^i)}^2(\mathbf{x},t,\tau ;\eta )\right]d\eta d\tau d\mathbf{x}}\hfill \\ {\displaystyle =-{\int }_{\Omega }{\int }_{{\alpha }_i}^{{\beta }_i}{\int }_0^{1}exp(-\eta c_i(t,\tau ))c_i(t,\tau )e_i^2(\mathbf{x},\tau ){(w^i)}^2(\mathbf{x},t,\tau ;\eta )d\eta d\tau d\mathbf{x}}\hfill \\ {\displaystyle -{\int }_{\Omega }{\int }_{{\alpha }_i}^{{\beta }_i}{e}_i^2(\mathbf{x},\tau )exp(-c_i(t,\tau ))\left(1-\frac{\partial c_i}{\partial t}(t,\tau )\right){(w^i)}^2(\mathbf{x},t,\tau ;1)d\tau d\mathbf{x}}\hfill \\ {\displaystyle +{\int }_{\Omega }\left({\int }_{{\alpha }_i}^{{\beta }_i}{e}_i^2(\mathbf{x},\tau )d\tau \right)\mid \frac{\partial u}{\partial t}{\mid }^2(\mathbf{x},t)d\mathbf{x}.}\hfill \end{array}$$

(77)

Introduce now the following functions:

$$\begin{array}{c}{\displaystyle \Phi (t)={\int }_{\Omega }u(\mathbf{x},t)\frac{\partial u}{\partial t}(\mathbf{x},t)d\mathbf{x},}\hfill \\ {\displaystyle \Psi (t)=\frac{\lambda }{\sqrt{\theta }}\sum _{j=1}^n{\int }_{\Omega }{\int }_0^1{\int }_{{\alpha }_j}^{{\beta }_j}{c}_j(t,\tau )exp(-\eta c_j(t,\tau ))e_j^2(\mathbf{x},\tau ){(w^j)}^2(\mathbf{x},t,\tau ;\eta )d\tau d\eta d\mathbf{x}.}\hfill \end{array}$$

(78)

From Equations (76) and (77) and Young’s inequality, we get

$$\begin{array}{c}{\displaystyle {\Phi }^{\prime }(t)=\Vert \frac{\partial u}{\partial t}{\Vert }_{L^2(\Omega )}^2-{\mathcal{K}(\Vert \sigma \nabla u\Vert }_{L^2(\Omega )}^2{)\Vert \sigma \nabla u\Vert }_{L^2(\Omega )}^2}\hfill \\ {\displaystyle -{\mathcal{M}(\Vert u\Vert }_{L^2(\Omega )}^2{)\Vert u\Vert }_{L^2(\Omega )}^2}\hfill \\ {\displaystyle -\zeta (t)\left(\sigma \nabla \frac{\partial u}{\partial t},\sigma \nabla u\right)-d_0(t){\int }_{\Omega }u(\mathbf{x},t)\frac{\partial u}{\partial t}(\mathbf{x},t)d\mathbf{x}}\hfill \\ {\displaystyle -\sum _{j=1}^n{\int }_{\Omega }{d}_j(t)\left({\int }_{{\alpha }_j}^{{\beta }_j}{e}_j^2(\mathbf{x},\tau )w^j(\mathbf{x},t,\tau ;1)u(\mathbf{x},t)d\tau \right)d\mathbf{x}}\hfill \\ {\displaystyle +{\int }_{\Omega }{u}^{2}ln\mid u{\mid }^{\theta }d\mathbf{x}}\hfill \\ {\displaystyle \le \left(1+\frac{{(C_{\Omega ,\sigma }{d}_0^{max})}^2}{2{\vartheta }_1{ϵ}_0}\right){\Vert \frac{\partial u}{\partial t}\Vert }_{L^2(\Omega )}^2-\zeta (t)\left(\sigma \nabla \frac{\partial u}{\partial t},\sigma \nabla u\right)}\hfill \\ {\displaystyle -\left(1-\frac{{ϵ}_0}{2}\right){\mathcal{K}(\Vert \sigma \nabla u\Vert }_{L^2(\Omega )}^2{)\Vert \sigma \nabla u\Vert }_{L^2(\Omega )}^2-{\mathcal{M}(\Vert u\Vert }_{L^2(\Omega )}^2{)\Vert u\Vert }_{L^2(\Omega )}^2}\hfill \\ {\displaystyle -\sum _{j=1}^n{\int }_{\Omega }{d}_j(t)\left({\int }_{{\alpha }_j}^{{\beta }_j}{e}_j^2(\mathbf{x},\tau )w^j(\mathbf{x},t,\tau ;1)u(\mathbf{x},t)d\tau \right)d\mathbf{x}+{\int }_{\Omega }{u}^{2}ln\mid u{\mid }^{\theta }d\mathbf{x}}\hfill \end{array}$$

and then

$$\begin{array}{c}{\displaystyle {\Phi }^{\prime }(t)\le \left(1+\frac{{(C_{\Omega ,\sigma }{d}_0^{max})}^2}{2{\vartheta }_1{ϵ}_0}\right)\Vert \frac{\partial u}{\partial t}{\Vert }_{L^2(\Omega )}^2-{\mathcal{M}(\Vert u\Vert }_{L^2(\Omega )}^2{)\Vert u\Vert }_{L^2(\Omega )}^2}\hfill \\ {\displaystyle -(1-\frac{{ϵ}_0+{ϵ}_1}{2}){\mathcal{K}(\Vert \sigma \nabla u\Vert }_{L^2(\Omega )}^2{)\Vert \sigma \nabla u\Vert }_{L^2(\Omega )}^2+\frac{{ϵ}_2\zeta (t)}{4}{\Vert \sigma \nabla u\Vert }_{L^2(\Omega )}^2}\hfill \\ {\displaystyle +\frac{{(C_{\Omega ,\sigma }{D}_{\infty })}^2{E}_{\infty }}{2{\vartheta }_1{ϵ}_1}\sum _{j=1}^n{\int }_{\Omega }{\int }_{{\alpha }_j}^{{\beta }_j}{e}_i^2(\mathbf{x},\tau ){(w^j)}^2(\mathbf{x},t,\tau ;1)d\tau d\mathbf{x}}\hfill \\ {\displaystyle +2\frac{\zeta (t)}{{ϵ}_2}{\Vert \sigma \nabla \frac{\partial u}{\partial t}\Vert }_{L^2(\Omega )}^2+{\int }_{\Omega }{u}^{2}ln\mid u{\mid }^{\theta }d\mathbf{x}.}\hfill \end{array}$$

(79)

where ${ϵ}_0,{ϵ}_1$ must be chosen appropriately, ${\displaystyle C_{\Omega ,\sigma }=C_{\Omega }{\underline{\varsigma }}^{-1}}$, ${\displaystyle E_{\infty }=\sum _{i=1}^n{\Vert e_i\Vert }_{L^2({\alpha }_i,{\beta }_i;L^{\infty }(\Omega ))}^2}$, ${\displaystyle D_{\infty }=\underset{i=1,n}{max}(d_i^{max})}$. Moreover, from Equation (77) and Young’s inequality, we can deduce

$$\begin{array}{c}{\displaystyle {\Psi }^{\prime }(t)\le -\frac{\lambda R_{1,\infty }}{\sqrt{\theta }}\sum _{j=1}^n{\int }_{\Omega }{\int }_{{\alpha }_j}^{{\beta }_j}{e}_j^2(\mathbf{x},\tau ){(w^j)}^2(\mathbf{x},t,\tau ;1)d\tau d\mathbf{x}+\frac{\lambda E_{\infty }}{\sqrt{\theta }}{\Vert \frac{\partial u}{\partial t}\Vert }_{L^2(\Omega )}^2}\hfill \\ {\displaystyle -\frac{\lambda R_{2,\infty }}{\sqrt{\theta }}\sum _{j=1}^n{\int }_{\Omega }{\int }_0^1{\int }_{{\alpha }_j}^{{\beta }_j}{c}_j(t,\tau )e_j^2(\mathbf{x},\tau ){(w^j)}^2(\mathbf{x},t,\tau ;\eta )d\tau d\eta d\mathbf{x},}\hfill \end{array}$$

(80)

where ${\displaystyle R_{1,\infty }=\underset{i=1,n}{min}\left(\underset{{\mathcal{T}}_i}{inf}\left(\left(1-\frac{\partial c_i}{\partial t}\right)exp(-c_i)\right)\right)}$ and ${\displaystyle R_{2,\infty }=\underset{i=1,n}{min}(exp(-\underset{{\mathcal{T}}_i}{sup}{c}_i))}$, with ${\mathcal{T}}_i=(0,+\infty )\times ({\alpha }_i,{\beta }_i)$.

Consequently

$$\begin{array}{c}{\displaystyle {(\Phi +\Psi )}^{\prime }(t)\le \left(1+\frac{{(C_{\Omega ,\sigma }{d}_0^{max})}^2}{2{\vartheta }_1{ϵ}_0}+\frac{\lambda E_{\infty }}{\sqrt{\theta }}\right){\Vert \frac{\partial u}{\partial t}\Vert }_{L^2(\Omega )}^2}\hfill \\ {\displaystyle -{\mathcal{M}(\Vert u\Vert }_{L^2(\Omega )}^2{)\Vert u\Vert }_{L^2(\Omega )}^2+{\int }_{\Omega }{u}^{2}ln\mid u{\mid }^{\theta }d\mathbf{x}}\hfill \\ {\displaystyle -\left(1-\frac{{ϵ}_0+{ϵ}_1}{2}\right){\mathcal{K}(\Vert \sigma \nabla u\Vert }_{L^2(\Omega )}^2{)\Vert \sigma \nabla u\Vert }_{L^2(\Omega )}^2}\hfill \\ {\displaystyle +\frac{{ϵ}_2\zeta (t)}{4}{\Vert \sigma \nabla u\Vert }_{L^2(\Omega )}^2+2\frac{\zeta (t)}{{ϵ}_2}{\Vert \sigma \nabla \frac{\partial u}{\partial t}\Vert }_{L^2(\Omega )}^2}\hfill \\ {\displaystyle -\left(\frac{\lambda R_{1,\infty }}{\sqrt{\theta }}-\frac{{(C_{\Omega ,\sigma }{D}_{\infty })}^2{E}_{\infty }}{2{\vartheta }_1{ϵ}_1}\right)\sum _{j=1}^n{\int }_{\Omega }{\int }_{{\alpha }_j}^{{\beta }_j}{e}_j^2(\mathbf{x},\tau ){(w^j)}^2(\mathbf{x},t,\tau ;1)d\tau d\mathbf{x}}\hfill \\ {\displaystyle -\frac{\lambda R_{2,\infty }}{\sqrt{\theta }}\sum _{j=1}^n{\int }_{\Omega }{\int }_0^1{\int }_{{\alpha }_j}^{{\beta }_j}{c}_j(t,\tau )e_j^2(\mathbf{x},\tau ){(w^j)}^2(\mathbf{x},t,\tau ;\eta )d\tau d\eta d\mathbf{x}.}\hfill \end{array}$$

(81)

Then, for all $ϵ,\xi >0$, we have (according to Equation (50) and by taking ${ϵ}_2=ϵ$)

$$\begin{array}{c}{\displaystyle \mathcal{Z}(t):={(ϵ(\Phi +\Psi )+\mathcal{E})}^{\prime }(t)+\frac{\lambda \xi }{\sqrt{\theta }}\mathcal{E}(t)}\hfill \\ {\displaystyle \le -\left(2{\aleph }_1(\lambda )-ϵ\left(1+\frac{{(C_{\Omega ,\sigma }{d}_0^{max})}^2}{2{\vartheta }_1{ϵ}_0}+\frac{\lambda E_{\infty }}{\sqrt{\theta }}\right)-\frac{\lambda \xi }{\sqrt{\theta }}\right){\Vert \frac{\partial u}{\partial t}\Vert }_{L^2(\Omega )}^2}\hfill \\ {\displaystyle -\frac{ϵ}{2}\left(2-\frac{ϵ\zeta (t)}{2{\vartheta }_1}-{ϵ}_0-{ϵ}_1\right){\mathcal{K}(\Vert \sigma \nabla u\Vert }_{L^2(\Omega )}^2{)\Vert \sigma \nabla u\Vert }_{L^2(\Omega )}^2}\hfill \\ {\displaystyle +\frac{\lambda \xi }{\sqrt{\theta }}{\mathcal{K}}_p{(\Vert \sigma \nabla u\Vert }_{L^2(\Omega )}^2{)-ϵ\mathcal{M}(\Vert u\Vert }_{L^2(\Omega )}^2{)\Vert u\Vert }_{L^2(\Omega )}^2}\hfill \\ {\displaystyle +\frac{\lambda \xi }{\sqrt{\theta }}{\mathcal{M}}_p{(\Vert u\Vert }_{L^2(\Omega )}^2)+\frac{\lambda \xi }{\sqrt{\theta }}\frac{\theta }{2}{\Vert u\Vert }_{L^2(\Omega )}^2+\left(ϵ-\frac{\lambda \xi }{\sqrt{\theta }}\right){\int }_{\Omega }{u}^{2}ln\mid u{\mid }^{\theta }d\mathbf{x}}\hfill \\ \\ {\displaystyle -\left(ϵ\left(\frac{\lambda R_{1,\infty }}{\sqrt{\theta }}-\frac{{(C_{\Omega ,\sigma }{D}_{\infty })}^2{E}_{\infty }}{2{\vartheta }_1{ϵ}_1}\right)+2{\aleph }_2(\lambda )\right)\sum _{j=1}^n{\int }_{\Omega }{\int }_{{\alpha }_j}^{{\beta }_j}{e}_j^2(\mathbf{x},\tau ){(w^j)}^2(\mathbf{x},t,\tau ;1)d\tau d\mathbf{x}}\hfill \\ {\displaystyle -\frac{\lambda }{\sqrt{\theta }}\left(ϵR_{2,\infty }-\xi \lambda \right)\sum _{j=1}^n{\int }_{\Omega }{\int }_0^1{\int }_{{\alpha }_j}^{{\beta }_j}{c}_j(t,\tau )e_j^2(\mathbf{x},\tau ){(w^j)}^2(\mathbf{x},t,\tau ;\eta )d\tau d\eta d\mathbf{x}.}\hfill \end{array}$$

(82)

Hence, from the logarithmic Sobolev inequality

$$\begin{array}{c}{\displaystyle \mathcal{Z}(t)\le -\left(2{\aleph }_1(\lambda )-ϵ\left(1+\frac{{(C_{\Omega ,\sigma }{d}_0^{max})}^2}{2{\vartheta }_1{ϵ}_0}+\frac{\lambda E_{\infty }}{\sqrt{\theta }}\right)-\rho \right){\Vert \frac{\partial u}{\partial t}\Vert }_{L^2(\Omega )}^2}\hfill \\ {\displaystyle -\frac{ϵ}{2}\left(2-\frac{ϵ\zeta (t)}{2{\vartheta }_1}-{ϵ}_0-{ϵ}_1\right){\mathcal{K}(\Vert \sigma \nabla u\Vert }_{L^2(\Omega )}^2{)\Vert \sigma \nabla u\Vert }_{L^2(\Omega )}^2+\rho {\mathcal{K}}_p{(\Vert \sigma \nabla u\Vert }_{L^2(\Omega )}^2)}\hfill \\ {\displaystyle -{ϵ\mathcal{M}(\Vert u\Vert }_{L^2(\Omega )}^2{)\Vert u\Vert }_{L^2(\Omega )}^2+\rho {\mathcal{M}}_p{(\Vert u\Vert }_{L^2(\Omega )}^2)}\hfill \\ {\displaystyle +\frac{\theta }{2}(ϵ-\rho )\left({ln\Vert u\Vert }_{L^2(\Omega )}^2-N(1+ln\alpha )\right){\Vert u\Vert }_{L^2(\Omega )}^2}\hfill \\ {\displaystyle +\left((ϵ-\rho )\frac{\theta {\alpha }^2}{2\pi {\underline{\varsigma }}^2}+\rho C_{\Omega ,\sigma }^2\frac{\theta }{2}\right){\Vert \sigma \nabla u\Vert }_{L^2(\Omega )}^2}\hfill \\ {\displaystyle -\left(ϵ\left(\frac{\lambda R_{1,\infty }}{\sqrt{\theta }}-\frac{{(C_{\Omega ,\sigma }{D}_{\infty })}^2{E}_{\infty }}{2{\vartheta }_1{ϵ}_1}\right)+2{\aleph }_2(\lambda )\right)\sum _{j=1}^n{\int }_{\Omega }{\int }_{{\alpha }_j}^{{\beta }_j}{e}_j^2(\mathbf{x},\tau ){(w^j)}^2(\mathbf{x},t,\tau ;1)d\tau d\mathbf{x}}\hfill \\ {\displaystyle -\frac{\lambda }{\sqrt{\theta }}\left(ϵR_{2,\infty }-\rho \sqrt{\theta }\right)\sum _{j=1}^n{\int }_{\Omega }{\int }_0^1{\int }_{{\alpha }_j}^{{\beta }_j}{c}_j(t,\tau )e_j^2(\mathbf{x},\tau ){(w^j)}^2(\mathbf{x},t,\tau ;\eta )d\tau d\eta d\mathbf{x},}\hfill \end{array}$$

with $\rho ={\displaystyle \frac{\lambda \xi }{\sqrt{\theta }}}$. Then (for ${\displaystyle \frac{ϵ}{2}}>\rho >0$ and $\varpi >0$)

$$\begin{array}{c}{\displaystyle \mathcal{Z}(t)\le -\left(2{\aleph }_1(\lambda )-ϵ\left(1+\frac{{(C_{\Omega ,\sigma }{d}_0^{max})}^2}{2{\vartheta }_1{ϵ}_0}+\frac{\lambda E_{\infty }}{\sqrt{\theta }}\right)-\rho \right){\Vert \frac{\partial u}{\partial t}\Vert }_{L^2(\Omega )}^2}\hfill \\ {\displaystyle -\left(\frac{ϵ}{2{\vartheta }_1}(2{\mu }_{\theta }-\frac{ϵ\zeta (t)}{2}-{ϵ}_3)+\rho -\frac{\rho C_{\Omega ,\sigma }^2}{{\vartheta }_1}(2{\mu }_{\theta }{\tilde{C}}_{\Omega ,\sigma }^2-\theta )\right){\mathcal{K}(\Vert \sigma \nabla u\Vert }_{L^2(\Omega )}^2{)\Vert \sigma \nabla u\Vert }_{L^2(\Omega )}^2}\hfill \\ {\displaystyle +\rho {\mathcal{K}}_p{(\Vert \sigma \nabla u\Vert }_{L^2(\Omega )}^2)}\hfill \\ {\displaystyle -{ϵ\mathcal{M}(\Vert u\Vert }_{L^2(\Omega )}^2{)\Vert u\Vert }_{L^2(\Omega )}^2+(ϵ-\rho )\frac{(1+\varpi )}{2}{\vartheta }_0{\Vert u\Vert }_{L^2(\Omega )}^2+\rho {\mathcal{M}}_p{(\Vert u\Vert }_{L^2(\Omega )}^2)}\hfill \\ {\displaystyle +\frac{\theta }{2}(ϵ-\rho )\left(g_{\frac{{\vartheta }_0}{\theta },\alpha }{(\Vert u\Vert }_{L^2(\Omega )})-\frac{\varpi {\vartheta }_0}{\theta }\right){\Vert u\Vert }_{L^2(\Omega )}^2}\hfill \\ {\displaystyle -\left(ϵ\left(\frac{\lambda P_{\infty }}{\sqrt{\theta }}-\frac{{(C_{\Omega ,\sigma }{D}_{\infty })}^2{E}_{\infty }}{2{\vartheta }_1{ϵ}_1}\right)+2{\aleph }_2(\lambda )\right)\sum _{j=1}^n{\int }_{\Omega }{\int }_{{\alpha }_j}^{{\beta }_j}{e}_j^2(\mathbf{x},\tau ){(w^j)}^2(\mathbf{x},t,\tau ;1)d\tau d\mathbf{x}}\hfill \\ {\displaystyle -\frac{\lambda }{\sqrt{\theta }}(ϵP_{\infty }-\rho \sqrt{\theta })\sum _{j=1}^n{\int }_{\Omega }{\int }_0^1{\int }_{{\alpha }_j}^{{\beta }_j}\tau e_j^2(\mathbf{x},\tau ){(w^j)}^2(\mathbf{x},t,\tau ;\eta )d\tau d\eta d\mathbf{x},}\hfill \end{array}$$

where ${\mu }_{\theta }={\vartheta }_1-{\displaystyle \frac{\theta {\alpha }^2}{2\pi {\underline{\varsigma }}^2}}>0$ (according to (H5)) and ${ϵ}_3=({ϵ}_0+{ϵ}_1){\vartheta }_1$. Consequently,

$${\displaystyle \begin{array}{c}{\displaystyle \mathcal{Z}(t)\le \frac{\theta }{2}(ϵ-\rho )\left(g_{\frac{{\vartheta }_0}{\theta },\alpha }{(\Vert u\Vert }_{L^2(\Omega )})-\varpi \frac{{\vartheta }_0}{\theta }\right){\Vert u\Vert }_{L^2(\Omega )}^2}\hfill \\ {\displaystyle -\left(2{\aleph }_1(\lambda )-ϵ(1+\frac{{(C_{\Omega ,\sigma }{d}_0^{max})}^2}{2{\vartheta }_1{ϵ}_0}+\frac{\lambda E_{\infty }}{\sqrt{\theta }})-\rho \right){\Vert \frac{\partial u}{\partial t}\Vert }_{L^2(\Omega )}^2}\hfill \\ {\displaystyle -\left(\frac{ϵ}{2{\vartheta }_1}(2{\mu }_{\theta }-{ϵ}_3)-\frac{\zeta (t)}{4{\vartheta }_1}{ϵ}^2-\frac{\rho C_{\Omega ,\sigma }^2}{{\vartheta }_1}(2{\mu }_{\theta }{\tilde{C}}_{\Omega ,\sigma }^2-\theta )\right){\mathcal{K}(\Vert \sigma \nabla u\Vert }_{L^2(\Omega )}^2{)\Vert \sigma \nabla u\Vert }_{L^2(\Omega )}^2}\hfill \\ {\displaystyle -\rho \left({\mathcal{K}(\Vert \sigma \nabla u\Vert }_{L^2(\Omega )}^2{)\Vert \sigma \nabla u\Vert }_{L^2(\Omega )}^2-{\mathcal{K}}_p{(\Vert \sigma \nabla u\Vert }_{L^2(\Omega )}^2\right)}\hfill \\ {\displaystyle -ϵ\left({\mathcal{M}(\Vert u\Vert }_{L^2(\Omega )}^2)-{\vartheta }_0\frac{(1+\varpi )}{2}\right){\Vert u\Vert }_{L^2(\Omega )}^2}\hfill \\ {\displaystyle -\rho \left({\mathcal{M}(\Vert u\Vert }_{L^2(\Omega )}^2{)\Vert u\Vert }_{L^2(\Omega )}^2-{\mathcal{M}}_p{(\Vert u\Vert }_{L^2(\Omega )}^2\right)}\hfill \\ {\displaystyle -\left(ϵ\left(\frac{\lambda R_{1,\infty }}{\sqrt{\theta }}-\frac{{(C_{\Omega ,\sigma }{D}_{\infty })}^2{E}_{\infty }}{2{\vartheta }_1{ϵ}_1}\right)+2{\aleph }_2(\lambda )\right)\sum _{j=1}^n{\int }_{\Omega }{\int }_{{\alpha }_j}^{{\beta }_j}{e}_j^2(\mathbf{x},\tau ){(w^j)}^2(\mathbf{x},t,\tau ;1)d\tau d\mathbf{x}}\hfill \\ {\displaystyle -\frac{\lambda }{\sqrt{\theta }}(ϵR_{2,\infty }-\rho \sqrt{\theta })\sum _{j=1}^n{\int }_{\Omega }{\int }_0^1{\int }_{{\alpha }_j}^{{\beta }_j}{c}_j(t,\tau )e_j^2(\mathbf{x},\tau ){(w^j)}^2(\mathbf{x},t,\tau ;\eta )d\tau d\eta d\mathbf{x},}\hfill \end{array}}$$

with a choice of ${ϵ}_0$ et ${ϵ}_1$ such that $2{\mu }_{\theta }-{ϵ}_3>0$.

For $ϵ$ and ${\displaystyle \frac{\rho }{ϵ}}$ sufficiently small, we have (according to the condition (75))

$$\begin{array}{c}{\displaystyle \mathcal{Z}(t)\le \frac{\theta }{2}(ϵ-\rho )\left(g_{\frac{{\vartheta }_0}{\theta },\alpha }{(\Vert u\Vert }_{L^2(\Omega )})-\frac{\varpi {\vartheta }_0}{\theta }\right){\Vert u\Vert }_{L^2(\Omega )}^2}\hfill \\ {\displaystyle -ϵ\left({\mathcal{M}(\Vert u\Vert }_{L^2(\Omega )}^2)-{\vartheta }_0\frac{(1+\varpi )}{2}\right){\Vert u\Vert }_{L^2(\Omega )}^2}\hfill \end{array}$$

By taking $\varpi =0$ if $E_a\le d_A$ and $\varpi ={\varpi }_a>0$ if $E_a\ge d_A$, we can deduce that

$$\begin{array}{c}{\displaystyle {(ϵ(\Phi +\Psi )+\mathcal{E})}^{\prime }(t)+\rho \mathcal{E}(t)=\mathcal{Z}(t)}\hfill \\ {\displaystyle \le \frac{\theta }{2}(ϵ-\rho )\left(g_{\frac{{\vartheta }_0}{\theta },\alpha }{(\Vert u\Vert }_{L^2(\Omega )})-\frac{\varpi {\vartheta }_0}{\theta }\right){\Vert u\Vert }_{L^2(\Omega )}^2.}\hfill \end{array}$$

(83)

Since $0<\mathcal{E}(0)<E_a\le d_I$, it follows from Equations (73) and (74) that, because $E_a\le d_A$ or ${\displaystyle 2{\mu }_{\theta }{\tilde{C}}_{\Omega ,\sigma }^2\le {\varpi }_a{\vartheta }_0}$, if $\varpi ={\varpi }_a>0$

$$\begin{array}{c}{\displaystyle {(ϵ(\Phi +\Psi )+\mathcal{E})}^{\prime }(t)\le -\rho \mathcal{E}(t).}\hfill \end{array}$$

(84)

To end the proof, we need to establish, for $ϵ$ sufficiently small, some energy equivalence relation. Since

$$\begin{array}{c}{\displaystyle \mid (\Phi +\Psi )(t)\mid \le {\int }_{\Omega }\mid u(\mathbf{x},t)\mid \mid \frac{\partial u}{\partial t}(\mathbf{x},t)\mid d\mathbf{x}}\hfill \\ {\displaystyle +\frac{\lambda }{\sqrt{\theta }}\sum _{i=1}^n{\int }_{\Omega }{\int }_0^1{\int }_{{\alpha }_i}^{{\beta }_i}{c}_i(t,\tau )e^{-\eta c_i(t,\tau )}{e}_i^2(\mathbf{x},\tau ){(w^i)}^2(\mathbf{x},t,\tau ;\eta )d\tau d\eta d\mathbf{x}}\hfill \\ {\displaystyle \le \frac{1}{\sqrt{\theta }}(\Vert \frac{\partial u}{\partial t}{\Vert }_{L^2(\Omega )}^2+\frac{\theta }{2}{\Vert u\Vert }_{L^2(\Omega )}^2}\hfill \\ {\displaystyle +\lambda \sum _{j=1}^n{\int }_{\Omega }{\int }_0^1{\int }_{{\alpha }_j}^{{\beta }_j}{c}_j(t,\tau )e_j^2(\mathbf{x},\tau ){(w^j)}^2(\mathbf{x},t,\tau ;\eta )d\tau d\eta d\mathbf{x})}\hfill \\ {\displaystyle \le \frac{1}{\sqrt{\theta }}\left(\mathcal{E}(t)-{\mathcal{K}}_p{(\Vert \sigma \nabla u\Vert }_{L^2(\Omega )}^2)-{\mathcal{M}}_p{(\Vert u\Vert }_{L^2(\Omega )}^2)+{\int }_{\Omega }{u}^{2}ln\mid u{\mid }^{\theta }d\mathbf{x}\right)}\hfill \\ {\displaystyle \le \frac{1}{\sqrt{\theta }}(\mathcal{E}(t)-\mathcal{I}(u))\le \frac{1}{\sqrt{\theta }}\mathcal{E}(t)(\sin \mathrm{ce}\mathcal{I}(u)\ge 0),}\hfill \end{array}$$

then we have the following energy equivalence

$${\displaystyle a_{ϵ}\mathcal{E}(t)\le (\mathcal{E}+ϵ(\Phi +\Psi ))(t)\le b_{ϵ}\mathcal{E}(t),}$$

(85)

where $a_{ϵ}=1-ϵ{\displaystyle \frac{1}{\sqrt{\theta }}}>0$ and $b_{ϵ}=1+ϵ{\displaystyle \frac{1}{\sqrt{\theta }}}>0$. From Equations (85) and (84), we can obtain that there exist constants ${\delta }_0$ and $C_0>0$ such that $(\mathcal{E}+ϵ(\Phi +\Psi ))(t)\le C_{0}exp(-{\delta }_{0}t)$ (where $C_0>0$ is constant depending on the initial data). According, again, to Equation (85), we can conclude the proof of the theorem. □

Remark 3.

If $\mathcal{K}$ and $\mathcal{M}$ are nondecreasing functions then Equation (75) is satisfied.

6. Conclusions

The main purpose of this paper is to study the existence, regularity and exponential stability of global solutions for the proposed class of nonlinear delayed Kirchhoff–Carrier wave-type models with different kinds of time-varying delays on bounded domains. These new nonlocal models include general mixed multiple time-varying delays, nonlocal operators, strong and weak damping cases, and nonlinear logarithmic source terms, as well as the nonlinear anisotropy diffusion and the anisotropic strong damping. The presence of nonlocal terms, anisotropic tensors and multiple time-varying delays leads to a more complex mathematical analysis of the dynamics of these derived models.

First, we reformulate the proposed nonlocal complex delay wave-type models into a nonlinear system coupling a nonlocal wave-type equation with ordinary differential equations, with the help of auxiliary functions. Due to the nature of the delay terms, nonlocal operators and the anisotropy of damping and diffusion, in order to analyze the problem in a rigorous manner, we need to show more regularity of the solution. Then, by applying logarithmic Sobolev inequality, Faedo–Galerkin’s method and some energy estimates, we established the local existence and some necessary regularity results for such a nonlocal and nonlinear system (with anisotropic diffusion and anisotropic damping). The global existence and energy decay rate of the solution are derived by virtue of the potential well method. Under some appropriate conditions, the stability through exponential decay of the global solution is established by constructing appropriate energy functionals and by applying the potential well and perturbed energy methods.

A future objective is to investigate the blow-up behavior of solutions of the considered problem. These studies will be the subject of a forthcoming paper. Moreover, the developed analysis in this work can be further applied to the study of the developed model as follows:

• With more complex delay functions as spatially anisotropic time delays or nonlocal time delays, i.e., by replacing in Equation (3) the term ${\displaystyle \frac{\partial u}{\partial t}}(\mathbf{x},t-c_j(t,\tau ))$ by ${\displaystyle \frac{\partial u}{\partial t}(\eta (\mathbf{x}),t-c_j(\mathbf{x},t,\tau ))}$ or by ${\int }_{\Omega }H\left(\mathbf{x},{\displaystyle \frac{\partial u}{\partial t}}(\mathbf{y},t-c_j(t,\tau ;\mathbf{x},\mathbf{y}))\right)d\mathbf{y}$ (where e.g., H is a Heaviside-type nonlinearity).
• Or by considering damping with memory as a Caputo-fractional damping model, i.e., by replacing $d_0(t){\displaystyle \frac{\partial u}{\partial t}}$ by ${\tilde{d}}_0(t){\partial }_{0^{+}}^{\alpha }u$, with $\alpha \in ]0,1]$ or by ${\tilde{d}}_0(t){\partial }_{0^{+}}^{p(\mathbf{x})}u$, where the function $p(\mathbf{x})\in ]0,1]$.

Finally, the proposed class of nonlocal wave-type equations involves several free parameters (or functions) and spatiotemporal kernels. It will be interesting to simulate and validate numerically the developed theoretical results by analyzing numerically the effects of these parameters or functions on the dynamics of these anisotropic history-dependent models.