\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2010(2010), No. 08, pp. 1--18.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2010 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2010/08\hfil Dissipative Boussinesq equations] {Dissipative Boussinesq equations on non-cylindrical domains in $\mathbb{R}^n$} \author[H. R. Clark, A. T. Cousin, C. L. Frota, J. L\'{\i}maco\hfil EJDE-2010/08\hfilneg] {Haroldo R. Clark, Alfredo T. Cousin, C\'{\i}cero L. Frota, Juan L\'{\i}maco} % in alphabetical order \address{Haroldo R. Clark \newline Universidade Federal Fluminense, IM-GAN, RJ, Brasil} \email{hclark@vm.uff.br} \address{Alfredo T. Cousin \newline Universidade Estadual de Maring\'a, DMA, PR, Brasil} \email{atcousin@uem.br} \address{C\'{\i}cero Lopes Frota \newline Universidade Estadual de Maring\'a, DMA, PR, Brasil} \email{clfrota@uem.br} \address{Juan L\'{\i}maco \newline Universidade Federal Fluminense, IM-GMA, RJ, Brasil} \email{jlimaco@vm.uff.br} \thanks{Submitted September 13, 2009. Published January 16, 2010.} \subjclass[2000]{35L10, 35Q53, 35B40} \keywords{Boussinesq equation; time dependent domains; existence; \hfill\break\indent uniqueness; asymptotic behavior} \begin{abstract} This article concerns the initial-boundary value problem for the nonlinear Boussinesq equations on time dependent domains in $\mathbb{R}^n$ with $1\leq n \leq 4$. Global solvability, uniqueness of solutions and the exponential decay to the energy are established provided the initial data are bounded in some sense. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \allowdisplaybreaks \section{Introduction} Let $\Omega \subset \mathbb{R}^n$ be an open bounded set with smooth boundary $\Gamma$. By $Q_{\infty} = \Omega \times (0,\infty)$ and $\Sigma_{\infty} = \Gamma \times (0,\infty)$ we denote the cylindrical domain and its boundary, respectively. Given $k = k(t)$ a real function defined on $[0,\infty)$, for each $t \geq 0$ we denote $\Omega_{t}$ the transformed sets by the number $k(t)$; that is, $$ \Omega_{t} = \{ x \in \mathbb{R}^n \text{ such that } x = k(t) y \text{ for all } y \in \Omega \}, $$ and $\Gamma_{t}$ is the boundary of $\Omega_{t}$. Then the time dependent domain $$ \widehat{\mathcal{Q}}_{\infty} = \cup_{t > 0} \big( \Omega_{t} \times \{ t \} \big), $$ is a subset of $\mathbb{R}^{n+1}$, with lateral boundary $$ {\widehat{\Sigma}_{\infty} = \cup_{t >0} \big( \Gamma_{t} \times \{ t \} \big).} $$ In this article, we study the initial-boundary value problem for the dissipative Boussinesq equation \begin{gather} u_{tt}(x,t)- \Delta \big(u(x,t)+u_t(x,t)+u^2(x,t)\big) +\Delta^{2} u(x,t)=0\quad\text{in } \widehat{\mathcal{Q}}_{\infty}, \label{1.1}\\ u = \frac{\partial u}{\partial \nu} = 0 \quad \text{on } \widehat{\Sigma}_{\infty}, \label{1.2}\\ u(x,0)=u_0(x); \; u_t(x,0)=u_1(x) \quad\text{for } x \in \Omega_0. \label{1.3} \end{gather} The theory of water waves for the case of shallow water and waves of small amplitude, idealized by Scott-Russell in 1834, had one of the first mathematical analysis established in 1872 by Boussinesq \cite{Boussinesq1}. His work derived a nonlinear dissipative wave system which is now known as the Boussinesq equations. See also Boussinesq \cite{Boussinesq2}. A nice survey on the history of the derivation of models of Boussinesq type can be found in Miles \cite{Miles}. Initial-boundary value problem in a cylindrical domain with small initial data has been considered by Varlamov \cite{Varlamov1,Varlamov2,Varlamov3} in both 1-dimensional and 2-dimensional cases. As results, classical solutions were constructed, uniqueness of solutions and the long-time asymptotic were obtained explicitly. For more information about problems associated with Boussinesq equation, see Varlamov \cite{Varlamov4} and references therein. Liu-Russell \cite{Liu-Russell} studied the existence and uniqueness of solutions to initial-boundary value problems on a 1-d periodic domain. There \eqref{1.1} have an internal weak damping $k_1 u_t$ and a linear feedback $k_2(u-[u])$. For the one-dimensional case, we mention the works of Bona-Sachs \cite{Bona-Sachs} and Tsutsumi-Matahshi \cite{Tsutsumi-Matahsshi}; where the authors studied the existence, uniqueness and stability of solutions for Cauchy problems. Cauchy problem related to \eqref{1.1} in an abstract framework on a Hilbert space $H$ has also been studied by other authors; Biler \cite{Biler} and Pereira \cite{Pereira} established results on existence, uniqueness and asymptotic stability of solutions. This article is motivated by the article \cite{Clark1} where a 1-d version of \eqref{1.1}-\eqref{1.3} is investigated. Our proof is a slight modification of the one in \cite{Clark1}. However we had to overcome some technically difficulties when considering this problem in $\widehat{\mathcal{Q}}_{\infty}$. The paper is organized as follows: In section 2, we give some assumptions to be used later, and state the main results. Subsequently, sections 3 and 4 are devoted to prove the main results: Theorems \ref{thm2.1} and \ref{thm2.2}. \section{Assumptions and main results} For the functional spaces we use standard notation as in Lions \cite{Lions1} and Lions-Magenes \cite{Lions-Magenes}. The inner product and norm in $L^{2}(\Omega)$ and $H_{0}^{1}(\Omega)$ are, respectively, denoted by \begin{gather*} (f,g) = \int_{\Omega} f(\xi)\, g(\xi) \,d\xi, \quad | f | = \Big(\int_{\Omega} | f(\xi) |^{2} \, d\xi \Big)^{1/2},\\ ((f,g)) = \sum_{i=1}^{n} \int_{\Omega} \frac{\partial f}{\partial \xi_{i}}(\xi) \frac{\partial g}{\partial \xi_{i}}(\xi) d\xi \,, \quad \| f \| = \Big( \sum_{i=1}^{n} \int_{\Omega} | \frac{\partial f}{\partial \xi_{i}}(\xi) |^{2} \, d\xi \Big)^{1/2}. \end{gather*} For the rest of this article we consider $n \leq 4$, which implies that $H_{0}^{1}(\Omega)$ is continuously imbedded in $L^{4}(\Omega)$. Let $C_{0}$ be such that $\| \cdot \|_{L^{4}(\Omega)} \leq C_{0}\| \cdot \|$. Moreover, let $C_{1}$ and $C_{2}$ be positive real constants satisfying the inequalities $\| f \|_{H^{2}(\Omega)}\leq C_{1} | \Delta f |$ and $\| f \| \leq C_{2} | \Delta f |$ for all $f \in H_{0}^{2}(\Omega)$. Since $\Omega$ is bounded there exists $C_{3}$ such that $| y_{i} | \leq \| y \|_{\mathbb{R}^n} \leq C_{3}$, for all $y = (y_{1}, \dots y_{n}) \in \Omega$. From Poincar\'{e} inequality $H_{0}^{1}(\Omega) \hookrightarrow L^{2}(\Omega)$ and we put $C_{4}$ such that $| \cdot | \leq C_{4} \| \cdot \|$. Henceforth we take for simplicity \begin{equation} C = \max_{0 \leq j \leq 4} C_{j} \,. \label{2.1} \end{equation} We now state some assumptions on the function $k$: \begin{gather} k \in C^{2}\big( [0,\infty) \big)\quad \text{with } k(0) = 1 \,, \label{2.2} \\ 0 < k_{0} \leq k(t) \leq k_{1} < \frac{1}{\sqrt 2\, C} \quad \text{for all } t \geq 0 \,. \label{2.3} \end{gather} Let $\epsilon_{0}$ be a real number such that \begin{equation} \epsilon_{0} > \frac{4}{1-4k_{1}^{2}C^{2}}\,, \label{2.4} \end{equation} and for each pair of functions $(u_{0},u_{1}) \in H_{0}^{2}(\Omega) \times L^{2}(\Omega)$ we denote \begin{equation} \alpha(u_{0}, u_{1}) = \frac{3}{4} | u_{1} |^{2} + \frac{1+C^{2}k_1^2}{k^2_{0}}\|u_0\|^2 + \frac{1}{2k_{0}^{4}} | \Delta u_{0} |^{2} \,. \label{2.5} \end{equation} We also introduce the following tree polynomials: \begin{gather} \begin{aligned} p(\lambda, \eta) &= \big[(2+9nC^{2}k_{1})C^{2}k_{1}+\frac{1}{k_0}\big] \lambda + \big[\frac{9}{4}\big((2n+1)^{2}+n^{4}C^{8}k_{1}^{2}\big)C^{4}\big] \lambda^{2} \\ &\quad + \big[ \frac{9}{2} n^{2}C^{4}\big] \lambda^{4} + \big[\frac{9}{8} n^{2}C^{4}k_{1}^{2}\big] \eta^{2} \,; \end{aligned} \label{2.6} \\ q(\lambda) = \frac{3}{k_{0}} \lambda ; \label{2.7} \\ r(\lambda , \eta) = \big[ \frac{2}{k_{0}} + C^{3} k_{1}^{3} (2n+n^{2}C^{2}) \big] \lambda+ (2nC^{4}k_{1}^{2})\lambda^2 + [ n C^{4} k_{1}^{3}] \eta \,.\label{2.8} \end{gather} Now that the notation and assumptions have been set, we state the main results. \begin{theorem}[Existence and exponential decay] \label{thm2.1} Suppose $n \leq 4$ and \eqref{2.2}--\eqref{2.3} hold. If \begin{equation} p( | k'(t) | , | k''(t) | ) < \frac{1}{4}\,, \quad q( | k'(t) | ) < \frac{1}{4}\,, \quad r( | k'(t) | , | k''(t) | ) < \frac{1}{4}\,, \label{2.9} \end{equation} for all $t \geq 0$. Then for each $(u_{0},u_{1}) \in H_{0}^{2}(\Omega) \times L^{2}(\Omega)$ such that \begin{equation} 2\epsilon_{0} C^{8} k_{1}^{6} \alpha(u_{0},u_{1}) + 8 C^{3} k_{1}^{2} \sqrt{\alpha (u_{0},u_{1})\,} < \frac{1}{4}\,, \label{2.10} \end{equation} there exists at least one global weak solution, $u$, to the problem \eqref{1.1}-\eqref{1.3}, such that \begin{equation} u \in L^{\infty}_{\rm loc}( 0, \infty ; H_{0}^{2}(\Omega_{t}) ), \quad u_{t} \in L^{2}_{\rm loc}( 0, \infty ; H^{1}_{0}(\Omega_{t}) ), \label{2.11} \end{equation} and it satisfies \eqref{1.1} in the sense of $L^2(0,T;H^{-2}(\Omega _t))$. Moreover, there exist positive real constants $\kappa_0, \kappa_1, \kappa_2$, such that the energy \[ E(u, t)=\frac{1}{2}\big\{|u'(t)|^2_{L^2(\Omega _t)} +|\nabla u(t)|^2_{L^2(\Omega _t)} + |\Delta u(t)|^2_{L^2(\Omega _t)} \big\} \] of system \eqref{1.1}-\eqref{1.3} satisfies \begin{equation} E(u,t)\leq \frac{\kappa_2\alpha (u_0, u_1)}{\kappa_1} \mathrm{e}^{-t/\kappa_0}\quad\text{ for all } t\geq 0, \label{2.12} \end{equation} where $\kappa_0,\kappa_1,\kappa_2$, are defined in \eqref{3.40}, \eqref{3.43}, \eqref{3.51}, respectively. \end{theorem} \begin{theorem}[Uniqueness of Solutions] \label{thm2.2} Under the assumption of Theorem \ref{thm2.1}, if $k'$ and $k''$ satisfy \begin{equation} |k'|_{L^1(0,+\infty)}+|k''|_{L^1(0,+\infty)} < \min\big\{\frac{1}{4K_1},\;\frac{1}{4K_2}\big\}, \label{2.13} \end{equation} where $K_1,K_2$ are real constants defined by \eqref{4.18}, then the global weak solution of \eqref{1.1}-\eqref{1.3} is unique on $[0, T]$, for all $T>0$. \end{theorem} \section{Proof of Theorem \ref{thm2.1}} The idea is to transform the non-cylindrical mixed problem \eqref{1.1}-\eqref{1.3} into to a problem on a cylindrical domain, by using a suitable change of variables. Whence let us introduce the function $F: \mathbb{R}^n \times [0,\infty) \to \mathbb{R}^n \times [0,\infty)$ defined by \begin{equation} F(y,t) = (k(t) y , t) = (k(t)y_{1}, \dots , k(t)y_{n},t)\,, \quad \text{for } y = (y_{1}, \dots, y_{n}) \in \mathbb{R}^n . \label{3.1} \end{equation} It is not difficult to see that $F$ is a diffeomorphism of class $C^{2}$ which satisfies: $$ F(Q_{\infty}) = \widehat{\mathcal{Q}}_{\infty},\quad F(\Omega) = \Omega_t,\quad F(\Sigma_{\infty}) = \widehat{\Sigma}_{\infty},\quad F^{-1}(x,t) = \big( \frac{x}{k(t)}, t\big) \,. $$ Given a function $u: \widehat{\mathcal{Q}}_{\infty} \to \mathbb{R}$, using the diffeomorphism $F$, we define $v = (u\circ F): Q_{\infty} \to \mathbb{R}$; that is, $v(y,t) = u(k(t)y , t)$. Then we get \begin{equation} \label{3.2} \begin{gathered} u(x,t) = v(y,t) \quad \text{where } y = \frac{x}{k(t)}\,,\\ \frac{\partial u}{\partial x_{i}} = \frac{1}{k(t)} \frac{\partial v}{\partial y_{i}} \quad \text{for } i = 1, \dots , n ,\\ \frac{\partial u}{\partial t} = - \frac{k'(t)}{k(t)} \sum_{j=1}^{n} \frac{\partial v}{\partial y_{j}}y_{j} + \frac{\partial v}{\partial t}\,. \end{gathered} \end{equation} For the second order derivatives we find \begin{equation} \label{3.3} \begin{gathered} \frac{\partial^{2} u}{\partial x_{i}^{2}} = \frac{1}{k^{2}(t)} \frac{\partial^{2} v}{\partial y_{i}^{2}} \quad \text{for } i = 1, \dots, n \,, \\ \begin{aligned} \frac{\partial^{2} u}{\partial t^{2}} &= \frac{\partial^{2} v}{\partial t^{2}} - 2 \frac{k'(t)}{k(t)} \sum_{j=1}^{n} \frac{\partial^{2} v}{\partial t \partial y_{j}} y_{j} + \big( \frac{k'(t)}{k(t)} \big)^{2} \sum_{j=1}^{n}\sum_{l=1}^{n} \frac{\partial^{2} v}{\partial y_{l} \partial y_{j}}\,y_{l} y_{j} \\ &\quad + \big[ \frac{2 (k'(t))^{2} - k(t) k''(t)}{k^{2}(t)} \big] \sum_{j=1}^{n} \frac{\partial v}{\partial y_{j}} y_{j}\,, \end{aligned} \\ \frac{\partial^{2} u}{\partial x_{i} \partial t} = - \frac{k'(t)}{k^{2}(t)} \sum_{j=1}^{n} \frac{\partial^{2} v}{\partial y_{i} \partial y_{j}} \,y_{j} - \frac{k'(t)}{k^{2}(t)} \frac{\partial v}{\partial y_{i}} + \frac{1}{k(t)} \frac{\partial^{2} v}{\partial y_{i} \partial t}\,, \\ \frac{\partial^{2}}{\partial x_{i}^{2}}( \frac{\partial u}{\partial t} ) = - \frac{k'(t)}{k^{3}(t)} \sum_{j=1}^{n} \frac{\partial^{3} v}{\partial y_{i}^{2} \partial y_{j}} y_{j} - 2 \frac{k'(t)}{k^{3}(t)} \frac{\partial^{2} v}{\partial y_{i}^{2}} + \frac{1}{k^{2}(t)} \frac{\partial^{3} v}{\partial y_{i}^{2} \partial t}\,. \end{gathered} \end{equation} Taking into account these computations, we have \begin{gather} \Delta u = \frac{1}{k^{2}(t)} \Delta v \,, \quad \Delta^{2} u = \frac{1}{k^{4}(t)} \Delta^{2} v\,,\label{3.4} \\ \Delta ( \frac{\partial u}{\partial t} ) = - \frac{k'(t)}{k^{3}(t)} \sum_{j=1}^{n} \Delta ( \frac{\partial v}{\partial y_{j}} ) y_{j} - 2 \frac{k'(t)}{k^{3}(t)} \Delta v + \frac{1}{k^{2}(t)} \Delta ( \frac{\partial v}{\partial t} ); \label{3.5} \\ \Delta (u^{2}) = \frac{1}{k^{2}(t)} \Delta (v^{2}). \label{3.6} \end{gather} From \eqref{3.3}-\eqref{3.6}, a function $u$ is a solution to the problem \eqref{1.1}-\eqref{1.3} if and only if $v$ is a solution to the problem \begin{gather} \begin{aligned} & v_{tt}(y,t)- \frac{1}{k^{2}(t)} \Delta ( v(y,t)+v_t(y,t)+v^2(y,t)) \frac{1}{k^{4}(t)} \Delta^{2} v(y,t)\\ &+ 2\frac{k'(t)}{k^{3}(t)} \Delta v(y,t) + \frac{k'(t)}{k^{3}(t)} \sum_{j=1}^{n} \Delta (\frac{\partial v}{\partial y_{j}}(y,t)) y_{j} - 2 \frac{k'(t)}{k(t)} \sum_{j=1}^{n} \frac{\partial^{2} v}{\partial t \partial y_{j}}(y,t) y_{j}\\ &+ \big(\frac{k'(t)}{k(t)}\big)^{2} \sum_{j=1}^{n} \sum_{l=1}^{n} \frac{\partial^{2} v}{\partial y_{l} \partial y_{j}}(y,t)y_{l} y_{j}\\ &+ \big[ \frac{2 (k'(t))^{2} - k(t) k''(t)}{k^{2}(t)} \big] \sum_{j=1}^{n} \frac{\partial v}{\partial y_{j}}(y,t) y_{j} = 0 \quad \text{in }Q_{\infty}, \end{aligned} \label{3.7} \\ v = \frac{\partial v}{\partial \nu} = 0 \quad \text{on }\Sigma_{\infty}, \label{3.8}\\ v(y,0)= v_{0}(y) = u_0(y), \quad v_t(y,0)=v_1(y)=u_{1}(y) \quad\text{for } y \in \Omega. \label{3.9} \end{gather} According to the above statements, it suffices to prove that under the assumptions of Theorem \ref{thm2.1} there exists at least a weak solution $v$ to \eqref{3.7}-\eqref{3.9} satisfying \begin{equation} v \in L^{\infty}_{\rm loc} ( 0, \infty ; H_{0}^{2}(\Omega)), \quad v_{t} \in L^{2}_{\rm loc}( 0, \infty ; H^{1}_{0}(\Omega) ). \label{3.10} \end{equation} Let $(w_j)_{j\in \mathbb N}$ be a basis to the Sobolev space $H_0^2(\Omega)$, and let $V_m$ be the finite dimensional subspace of $H_0^2(\Omega)$ spanned by the vectors $\{ w_1, w_2, \dots , w_m \}$. The theory of ordinary differential equations yields a local solution $ v_{m}(x,t)=\sum_{j=1}^m g_{jm}(t) w_j(x)$ in $V_{m}$, defined in $[0,T_{m}]$ for each $m \in {\mathbb N}$. This solution is a local solution to the approximate initial value problem \begin{gather} \begin{aligned} &(v''_{m}(t), w) + \frac{1}{k^{2}(t)} \Big( \nabla ( v_{m}(t)+v_{m}'(t) +v_{m}^2(t)) , \nabla w \Big) + \frac{1}{k^{4}(t)} (\Delta v_{m}(t),\Delta w) \\ &- 2 \frac{k'(t)}{k^{3}(t)} \Big( \nabla v_{m}(t), \nabla w \Big) - \frac{k'(t)}{k^{3}(t)} \sum_{j=1}^{n} \Big(\nabla (\frac{\partial v_{m}}{\partial y_{j}}(t)), \nabla (y_{j} w)\Big) \\ & - 2 \frac{k'(t)}{k(t)} \sum_{j=1}^{n} \Big(\frac{\partial v_{m}'}{\partial y_{j}}(t) y_{j}, w\Big) + ( \frac{k'(t)}{k(t)})^{2} \sum_{i=1}^{n} \sum_{j=1}^{n} \Big(\frac{\partial^{2} v_{m}}{\partial y_{i} \partial y_{j}}(t)y_{i}y_{j}\, , w\Big) \\ & + \big[ \frac{2 (k'(t))^{2} - k(t) k''(t)}{k^{2}(t)} \big] \sum_{j=1}^{n} \big(\frac{\partial v_{m}}{\partial y_{j}}(t) y_{j}, w\big) = 0\quad \text{for all } w \in V_{m}\,, \end{aligned}\label{3.11} \\ v_{m}(0)= v_{0m} \to u_{0} \quad \text{in } H_{0}^{2}(\Omega)\quad \text{and} \quad v_{m}'(0)= v_{1 m} \to u_{1} \quad \text{in } L^{2}(\Omega). \label{3.12} \end{gather} Now we need estimates independent of $m$ and $t$ which will allow us to extend the solutions $v_{m}$ to the whole interval $[0,\infty)$ and take to the limit in $v_{m}$ as $m \to \infty$. \subsection*{A priori estimates} First we take $w = v_{m}'$ in (\ref{3.11}) to obtain \begin{align} & \frac{1}{2} \frac{d}{dt} | v_{m}'(t) | ^{2} + \frac{1}{ k^{2}(t)} \frac{d}{dt} \| v_{m}(t) \|^{2}+ \frac{1}{k^{2}(t)} \| v_{m}'(t) \|^{2} + \frac{1}{k^{2}(t)} (\nabla v_{m}^{2}(t) , \nabla v_{m}'(t) ) \nonumber\\ &+ \frac{1}{2k^{4}(t)} \frac{d}{dt} | \Delta v_{m}(t) |^{2} - \frac{2k'(t)}{k^{3}(t)}(\nabla v_m(t), \nabla v_m'(t)) \nonumber\\ &- \frac{k'(t)}{k^{3}(t)} \sum_{j=1}^{n} \Big(\nabla (\frac{\partial v_{m}}{\partial y_{j}}(t)), \nabla (y_{j} v_{m}'(t) )\Big) \label{3.13} \\ & - 2 \frac{k'(t)}{k(t)} \sum_{j=1}^{n} \Big(\frac{\partial v_{m}'}{\partial y_{j}}(t) y_{j}, v_{m}'(t) \Big) + ( \frac{k'(t)}{k(t)})^{2} \sum_{i=1}^{n} \sum_{j=1}^{n} \Big(\frac{\partial^{2} v_{m}}{\partial y_{i} \partial y_{j}}(t)y_{i}y_{j}\, , v_{m}'(t) \Big) \nonumber\\ &+ \big[ \frac{2 (k'(t))^{2} - k(t) k''(t)}{k^{2}(t)} \big] \sum_{j=1}^{n} \Big(\frac{\partial v_{m}}{\partial y_{j}}(t) y_{j}, v_{m}'(t)\Big) = 0 . \nonumber \end{align} Now we study each term in (\ref{3.13}): \begin{equation} \label{3.14} \frac{1}{k^{2}(t)} \frac{d}{dt} \| v_{m}(t) \|^{2} \\ = \frac{d}{dt} (\frac 12 \frac{\| v_m(t) \|^2}{k^2(t)}) +\frac{k'(t)}{k^3(t)}\|v_m(t)\|^2; \end{equation} \begin{equation} \label{3.15} \begin{aligned} &\big| \frac{1}{k^{2}(t)} (\nabla v_{m}^{2}(t) , \nabla v_{m}'(t) ) \big|_\mathbb{R} \\ &\leq \frac{2}{k^{2}(t)}\sum_{i=1}^{n} \| v_{m}(t) \|_{L^{4}(\Omega)} \, \| \frac{\partial v_{m}}{\partial y_{i}}(t) \|_{L^{4}(\Omega)} \, | \frac{\partial v_{m}'}{\partial y_{i}}(t) | \\ & \leq \frac{2 C^{2}}{k^{2}(t)} \| v_{m}(t) \| \sum_{i=1}^{n} \| \frac{\partial v_{m}}{\partial y_{i}}(t) \| \, | \frac{\partial v_{m}'}{\partial y_{i}}(t) | \\ &\leq \frac{2 C^{2}}{k^{2}(t)} \| v_{m}(t) \| \, \| v_{m}(t) \|_{H^{2}(\Omega)} \, \| v_{m}'(t) \| \\ & \leq \frac{2 C^{4}}{k^{2}(t)} | \Delta v_{m}(t) |^{2} \, \| v_{m}'(t) \| \\ & \leq \epsilon_{0} C^{8} \frac{| \Delta v_{m}(t) |^{4}}{k^{2}(t)} + \frac{1}{\epsilon_{0}} \frac{\| v_{m}'(t) \|^{2}}{k^{2}(t)} \, ; \end{aligned} \end{equation} here we have used the imbedding $H_{0}^{1}(\Omega) \hookrightarrow L^{4}(\Omega)$ and the constants $C$ and $\epsilon_{0}$ given in \eqref{2.1} and \eqref{2.4}, respectively. We also find \begin{gather} \frac{1}{2k^{4}(t)} \frac{d}{dt} | \Delta v_{m}(t) |^{2} = \frac{d}{dt} \Big( \frac{1}{2k^{4}(t)} | \Delta v_{m}(t) |^{2} \Big) + \frac{2 k'(t)}{k^{5}(t)} | \Delta v_{m}(t) |^{2}\,; \label{3.16} \\ \big|\frac{2k'(t)}{k^3(t)}(\nabla v_m(t),\nabla v_m(t))\big|_\mathbb{R} \leq \frac{|k'(t)|}{k_0}\frac{\| v_m(t)\|^2}{k^2(t)} +\frac{|k'(t)|}{k_0}\frac{\| v_m'(t)\|^2}{k^2(t)}. \label{3.17} \end{gather} Taking $\delta_{i}^{j} = 0$ if $i=j$ and 1 if $i \neq j$, we have \begin{equation} \begin{aligned} &\big| \frac{k'(t)}{k^{3}(t)} \sum_{j=1}^{n} \Big(\nabla (\frac{\partial v_{m}}{\partial y_{j}}(t)), \nabla (y_{j}\, v_{m}'(t) )\Big) \big|_\mathbb{R}\\ &= \Big| \frac{k'(t)}{k^{3}(t)}\Big[ \sum_{i=1}^{n} \Big( \frac{\partial^{2} v_{m}}{\partial y_{i}^{2}}(t) , v_{m}'(t) \Big) + \sum_{i=1}^{n} \Big( \frac{\partial^{2} v_{m}}{\partial y_{i}^{2}}(t) , y_{i} \frac{\partial v_{m}'}{\partial y_{i}}(t) \Big) \\ &\quad + \sum_{j=1}^{n} \sum_{i=1}^{n} \delta_{i}^{j} \Big( \frac{\partial^{2} v_{m}}{\partial y_{i} \partial y_{j} }(t) , y_{j} \frac{\partial v_{m}'}{\partial y_{i}}(t) \Big) \Big] \Big|_\mathbb{R} \\ &\leq \frac{| k'(t) | }{k^{3}(t) } (2n+1) C^{2} \| v_{m}'(t) \| \, | \Delta v_{m}(t) | \\ &\leq \frac{9 (2n+1)^{2} C^{4}}{4} | k'(t) |^{2} \frac{\| v_{m}'(t) \|^{2}}{k^{2}(t)} + \frac{1}{9} \frac{| \Delta v_{m}(t) |^2}{k^{4}(t)}\,; \end{aligned} \label{3.18} \end{equation} \begin{equation} \begin{aligned} \Big| 2 \frac{k'(t)}{k(t)} \sum_{j=1}^{n} \Big(\frac{\partial v_{m}'}{\partial y_{j}}(t) y_{j}, v_{m}'(t) \Big) \Big|_\mathbb{R} &\leq 2 C \frac{| k'(t)| }{k(t)} \sum_{j=1}^{n} | \frac{\partial v_{m}'}{\partial y_{j}}(t) | \, | v_{m}'(t) | \\ &\leq 2 C \frac{| k'(t)| }{k(t)} | v_{m}'(t) | \, \| v_{m}'(t) \| \\ &\leq 2 C^{2} \frac{| k'(t)| }{k(t)} \| v_{m}'(t) \|^{2}\,; \end{aligned} \label{3.19} \end{equation} \begin{equation} \begin{aligned} &\Big| ( \frac{k'(t)}{k(t)})^{2} \sum_{i=1}^{n} \sum_{j=1}^{n} \Big(\frac{\partial^{2} v_{m}}{\partial y_{i} \partial y_{j}}(t)y_{i}\,y_{j}\, , v_{m}'(t) \Big) \Big|_\mathbb{R}\\ & \leq C^{2} \frac{| k'(t) |^{2}}{k^{2}(t)} | v_{m}'(t) | \sum_{i=1}^{n} \sum_{j=1}^{n} | \frac{\partial^{2} v_{m}}{\partial y_{i} \partial y_{j}}(t) |\\ &\leq n^{2} C^{4} \frac{| k'(t) |^{2}}{k^{2}(t)} \| v_{m}'(t) \| \, | \Delta v_{m}(t) |\\ & \leq \frac{ 9 n^{4} C^{8}}{4} | k'(t) |^{2} \| v_{m}'(t) \|^{2} + \frac{1}{9} \frac{| \Delta v_{m}(t) |^{2}}{k^{4}(t) } \,; \end{aligned}\label{3.20} \end{equation} \begin{equation} \label{3.21} \begin{aligned} &\Big| \big[ \frac{2 (k'(t))^{2} - k(t) k''(t)}{k^{2}(t)} \big] \sum_{j=1}^{n} \Big(\frac{\partial v_{m}}{\partial y_{j}}(t) y_{j}, v_{m}'(t)\Big) \Big|_\mathbb{R} \\ & \leq C^{2} n \frac{| 2 (k'(t))^{2} - k(t) k''(t)| }{k(t)} \, \| v_{m}'(t) \| \,\frac{\| v_{m}(t) \|}{k(t)} \\ &\leq \frac{9 C^{4} n^{2}}{4} | 2 (k'(t))^{2} - k(t) k''(t)|^{2} \frac{\| v_{m}'(t) \|^{2}}{k^{2}(t)} + \frac{1}{9} \frac{\| v_{m}(t) \|^{2}}{k^{2}(t)}\,; \end{aligned} \end{equation} Inserting (\ref{3.14})-(\ref{3.21}) in (\ref{3.13}) we obtain \begin{equation} \begin{aligned} & \frac{1}{2} \frac{d}{dt} \big[ | v_{m}'(t) |^{2} + \frac{\| v_{m}(t)\|^{2}}{k^{2}(t)} + \frac{| \Delta v_{m}(t) |^{2}}{k^{4}(t)} \big] + \frac{\| v_{m}' \|^{2}}{k^{2}(t)} \\ & \leq \big[ \frac{1}{9} + \frac{2}{k_{0}} | k'(t) | \big] \frac{\| v_{m}(t) \|^{2}}{k^{2}(t)} + \Big[ \frac{1}{\epsilon_{0}} + (2 C^{2} k_{1}+\frac{1}{k_0})| k'(t) |\\ &\quad + \frac{9 C^{4} \big( (2n+1)^{2} + C^{8} n^{4} k_{1}^{2} \big) }{4} | k'(t) |^{2} + \frac{9 C^{4} n^{2}}{2} | k'(t) |^{4} \\ &\quad +\frac{9 C^{4} n^{2} k_{1}^{2}}{8} | k''(t) |^{2} \Big] \frac{\| v_m'(t) \|^{2}}{k^{2}(t)} + \Big( \frac{2}{9} + \frac{2}{k_{0}} | k'(t)| \Big) \frac{| \Delta v_m(t) |^{2}}{k^{4}(t)}\\ &\quad + \epsilon_{0} C^{8} \frac{| \Delta v_m(t) |^{4}}{k^{2}(t)}\,. \end{aligned}\label{3.22} \end{equation} Now we go back to (\ref{3.11}) and take $w = v_m(t) $. Hence \begin{align} &\frac{d}{dt} (v_m'(t) , v_m(t)) - | v_m'(t) |^{2} + \frac{\| v_m(t) \|^{2}}{k^{2}(t)} + \frac{1}{2k^{2}(t)} \frac{d}{dt} \| v_m(t) \|^{2} \nonumber\\ &+\frac{1}{k^{2}(t)} \Big( \nabla (v_m(t))^{2} , \nabla v_m(t) \Big) + \frac{| \Delta v_m(t) |^{2}}{k^{4}(t)} - 2 \frac{k'(t)}{k^{3}(t)} \| v_m(t) \|^{2} \nonumber\\ & - \frac{k'(t)}{k^{3}(t)} \sum_{j=1}^{n} \Big( \nabla(\frac{\partial v_{m}}{\partial y_{j}}(t)) , \nabla (y_{j}\, v_m(t)) \Big) - \frac{2 k'(t)}{k(t)} \sum_{j=1}^{n} \Big( \frac{\partial v_{m}'}{\partial y_{j}}(t)y_{j} , v_m(t) \Big) \nonumber\\ &+ \frac{k'(t)}{k(t)} \sum_{i=1}^{n}\sum_{j=1}^{n} \Big( \frac{\partial^{2} v_{m}}{\partial y_{i} \partial y_{j}}(t)y_{i} y_{j} , v_m(t) \Big) \label{3.23} \\ &+ \Big( \frac{2 (k'(t))^{2} - k(t) k''(t)}{k^{2}(t)} \Big) \sum_{j=1}^{n} \Big( \frac{\partial v_{m}}{\partial y_{j}}(t)y_{j} , v_m(t) \Big) = 0 \,. \nonumber \end{align} We work with each term of (\ref{3.23}): \begin{equation} \frac{1}{2k^{2}(t)} \frac{d}{dt} \| v_m(t) \|^{2} = \frac{d}{dt} \left[ \frac{1}{2k^{2}(t)} \| v_m(t) \|^{2} \right] + \frac{k'(t)}{k^{3}(t)} \| v_m(t) \|^{2} \,;\label{3.24} \end{equation} \begin{equation} \begin{aligned} &\Big| \frac{1}{k^{2}(t)} \Big( \nabla (v_m(t))^{2} , \nabla v_m(t) \Big) \Big|_\mathbb{R} \\ &\leq \frac{2}{k^{2}(t)} \sum_{i=1}^{n} \int _{\Omega} \big| v_{m}(x,t) \big|_\mathbb{R} \big| \frac{\partial v_{m}}{\partial y_{i}}(x,t) \big|_\mathbb{R} \big| \frac{\partial v_{m}}{\partial y_{i}}(x,t) \big|_\mathbb{R} \,dy \\ &\leq \frac{2}{k^{2}(t)} \sum_{i=1}^{n} \| v_m(t) \|_{L^{4}(\Omega)} \| \frac{\partial v_{m}}{\partial y_{i}}(t) \|_{L^{4}(\Omega)} | \frac{\partial v_{m}}{\partial y_{i}}(t) |\\ & \leq \frac{2 C^{2} n}{k^{2}(t)} \| v_m(t) \|^{2} \, \| v_m(t) \|_{H^{2}(\Omega)} \\ &\leq \frac{2 n C^{3}}{k^{2}(t)} \, \| v_m(t) \| \, | \Delta v_m(t) |^{2}\,; \end{aligned}\label{3.25} \end{equation} \begin{equation} \begin{aligned} &\big| \frac{k'(t)}{k^{3}(t)} \sum_{j=1}^{n} \Big( \nabla(\frac{\partial v_{m}}{\partial y_{j}}(t)) , \nabla (y_{j}\, v_m(t)) \Big) \big|_\mathbb{R} \\ &\leq \frac{| k'(t)| }{k^{3}(t)}\Big[ | v_m(t) | \sum_{i=1}^{n} | \frac{\partial^{2} v_{m}}{\partial y_{i}^{2}}(t) | + C \sum_{j=1}^{n} \sum_{i=1}^{n} | \frac{\partial^{2} v_{m}}{\partial y_{i} \partial y_{j}}(t) | | \frac{\partial v_{m}}{\partial y_{i}}(t) | \Big] \\ &\leq \frac{| k'(t)| }{k^{3}(t)} [ n | v_m(t) | \| v_m(t) \|_{H^{2}(\Omega)} + n C \| v_m(t) \| \| v_m(t) \|_{H^{2}(\Omega)}]\\ & \leq 2 n C^{3} \frac{| k'(t)| }{k^{3}(t)} | \Delta v_m(t) |^{2}\,; \end{aligned} \label{3.26} \end{equation} \begin{equation} \begin{aligned} \Big| \frac{2 k'(t)}{k(t)} \sum_{j=1}^{n} \Big( \frac{\partial v_{m}'}{\partial y_{j}}(t)y_{j} , v_m(t) \Big)\Big|_\mathbb{R} & \leq \frac{2 C | k'(t)|}{k(t)} \sum_{j=1}^{n} | \frac{\partial v_{m}'}{\partial y_{j}}| | v_m(t) | \\ &\leq 2 \frac{\| v_m(t) \|}{k(t)} \, nC^{2} | k'(t) | \| v_m'(t) \|\\ &\leq 9 n C^{4} | k'(t) |^{2} \| v_m'(t) \|^{2} + \frac{1}{9} \frac{\| v_m(t) \|^{2}}{k^{2}(t)} \, ; \end{aligned} \label{3.27} \end{equation} \begin{equation} \begin{aligned} &\Big| \Big( \frac{2 (k'(t))^{2} - k(t) k''(t)}{k^{2}(t)} \Big) \sum_{j=1}^{n} \Big( \frac{\partial v_{m}}{\partial y_{j}}(t)y_{j} , v_m(t) \Big) \Big|_\mathbb{R} \\ &\leq \Big( \frac{2 | k'(t) |^{2} + k(t) | k''(t)| }{k^{2}(t)} \Big) C | v_m(t) | \sum_{j=1}^{n} | \frac{\partial v_{m}}{\partial y_{j}}(t) | \\ &\leq \Big( \frac{2 | k'(t) |^{2} + k(t) | k''(t)| }{k^{2}(t)} \Big) n C^{4} | \Delta v_m(t) |^{2} \,. \end{aligned} \label{3.28} \end{equation} Since $0 < k_{0} \leq k(t) \leq k_{1}$, taking into account (\ref{3.23})-(\ref{3.28}) we find \begin{equation} \begin{aligned} &\frac{d}{dt} \big[ (v_m'(t) , v_m(t) ) + \frac{1}{2} \frac{\| v_m(t) \|^{2}}{k^{2}(t)} \big] + \frac{8}{9} \frac{\| v_m(t) \|^{2}}{k^{2}(t)} + \frac{| \Delta v_m(t) |^{2}}{k^{4}(t)} \\ & \leq \frac{1}{k_{0}} | k'(t) | \frac{\| v_m(t) \|^{2}}{k^{2}(t)} + \Big( k_{1}^{2} C^{2} + 9 n C^{4} k_{1}^{2} | k'(t) | \Big) \frac{\| v_m'(t) \|^{2}}{k^{2}(t)} \\ &\quad + 2nC^{3} k_{1}^{2} \| v_m(t) \| \frac{| \Delta v_m(t) |^{2}}{k^{4}(t)} + \Big[ (2n+n^{2} C^{2}) C^{3} k_{1}^{3} | k'(t) | \\ &\quad + n C^{4} k_{1}^{2} \Big( 2 | k'(t) |^2 + k_{1} | k''(t) | \Big) \Big] \frac{| \Delta v_m(t) |^{2}}{k^{4}(t)} \,. \end{aligned} \label{3.29} \end{equation} This inequality and (\ref{3.22}) yields \begin{equation} \begin{aligned} &\frac{dH}{dt}(t)+\big(1 - k_{1}^{2}C^{2}-\frac{1}{\epsilon_{0}}\big) \frac{\| v_m'(t) \|^{2}}{k^{2}(t)} + \frac{7}{9} \frac{\| v_m(t) \|^{2}}{k^{2}(t)} + \frac{7}{9} \frac{| \Delta v_m(t) |^{2}}{k^{4}(t)} \\ &\leq p( | k'(t) | , | k''(t) | ) \frac{\| v_m'(t) \|^{2}}{k^{2}(t)}+ q( | k'(t) | ) \frac{\| v_m(t) \|^{2}}{k^{2}(t)}\\ &\quad + r( | k'(t) | , | k''(t) | ) \frac{| \Delta v_m(t) |^{2}}{k^{4}(t)} + \epsilon_{0} C^{8} \frac{| \Delta v_m(t) |^{4}}{k^{2}(t)}\\ &\quad + 2nC^{3}k_{1}^{2} \| v_m(t) \| \frac{| \Delta v_m(t) |^{2}}{k^{4}(t)}\, , \end{aligned} \label{3.30} \end{equation} where \begin{equation} H(t) = \frac{1}{2} | v_m'(t) |^{2} + \frac{\| v_m(t) \|^{2}}{k^{2}(t)} + \frac{1}{2} \frac{| \Delta v_m(t) |^{2}}{k^{4}(t)} + (v_m'(t), v_m(t)). \label{3.31} \end{equation} From \eqref{2.3} and (\ref{2.4}) we can see that $(1 - k_{1}^{2}C^{2}-\frac{1}{\epsilon_{0}})> 3/4$. Therefore, we rewrite (\ref{3.30}) as \begin{equation} \begin{aligned} &\frac{dH}{dt}(t)+ \big( \frac{3}{4} - p( | k'(t) | , | k''(t) | )\big) \frac{\| v_m'(t) \|^{2}}{k^{2}(t)} \\ &+ \big( \frac{3}{4} - q( | k'(t) | ) \big) \frac{\| v_m(t) \|^{2}}{k^{2}(t)} + ( \frac{3}{4} - r( | k'(t) | , | k''(t) | ) ) \frac{| \Delta v_m(t) |^{2}}{k^{4}(t)}\\ & - \epsilon_{0} C^{8} \frac{| \Delta v_m(t) |^{4}}{k^{2}(t)} - 2nC^{3}k_{1}^{2} \| v_m(t) \| \frac{| \Delta v_m(t) |^{2}}{k^{4}(t)}\leq 0\,. \end{aligned}\label{3.32} \end{equation} This inequality and (\ref{2.9}) yield \begin{equation} \frac{dH}{dt}(t)+ \frac{1}{2} \frac{\| v_m'(t) \|^{2}}{k^{2}(t)} + \frac{1}{2} \frac{\| v_m(t) \|^{2}}{k^{2}(t)} + \frac{1}{4} \frac{| \Delta v_m(t) |^{2}}{k^{4}(t)} + \Big( \frac{1}{4} - \gamma (t) \Big) \frac{| \Delta v_m(t) |^{2}}{k^{4}(t)}\leq 0\, ,\label{3.33} \end{equation} where $$ \gamma (t) = \epsilon_{0} C^{8} k_{1}^{2} | \Delta v_m(t) |^{2} + 2 n C^{3} k_{1} \| v_m(t) \|\,. $$ On the other hand, \begin{equation} | ( v_m'(t) , v_m(t) ) | \leq \frac{1}{4} | v_m'(t) |^{2} + C^{2} k_{1}^{2}\frac{\| v_m(t) \|^{2}}{k^{2}(t)}\,. \label{3.34} \end{equation} Taking into account the definition of $H(t)$, (\ref{3.33}) and (\ref{3.34}), for all $t \geq 0$ we find \begin{equation} \begin{aligned} &\frac{1}{4}| v_m'(t) |^2 +\frac 12\frac{\|v_m(t)\|^2}{k^2(t)}+\frac 12\frac{|\Delta v_m(t) |^2}{k^4(t)}\\ &\leq H(t) \\ & \leq \frac{3}{4} | v_m'(t) |^{2} + \frac{(1+C^2k_1^2)}{k_0^2} \| v_m(t) \|^{2} + \frac{1}{2k_{0}^{4}} | \Delta v_m(t) |^{2} \,, \end{aligned} \label{3.35} \end{equation} which in particular for $t=0$ gives $H(0) \leq \alpha (u_{0},u_{1})$. Simple computations then lead to \begin{equation} \gamma (t) \leq 2\epsilon_{0} C^{8} k_{1}^{6} H(t) + 2 n C^{3} k_{1}^{2} H^{{1}/{2}}(t) \quad \forall t \geq 0\,, \label{3.36} \end{equation} and from (\ref{2.10}), we obtain $\gamma (0) \leq 2 \epsilon_{0} C^{8} k_{1}^{6} \alpha (u_{0},u_{1}) + 2 n C^{3} k_{1}^{2} \sqrt{\alpha (u_{0},u_{1})\,} < {1}/{4}$. Now we claim that \begin{equation} \gamma (t) < \frac{1}{4} \quad \text{for all } t \geq 0. \label{3.37} \end{equation} By contradiction let us suppose that (\ref{3.37}) does not hold. The continuity gives $t^{*} > 0$ such that \begin{equation} \gamma (t) < \frac{1}{4} \quad \text{for all } t \in [0,t^{*}) \quad \text{and} \quad \gamma (t^{*}) = \frac{1}{4}\,. \label{3.38} \end{equation} Integrating (\ref{3.32}) from 0 to $t^{*}$ we come to $ H(t^{*}) \leq H(0) \leq \alpha (u_{0},u_{1})$. This inequality, (\ref{3.36}) and (\ref{2.10}) yield ${\gamma (t^{*}) < {1}/{4}}$, which contradicts (\ref{3.38}) and our claim is proved. Since we have (\ref{3.32}), (\ref{3.35}) and (\ref{3.37}) one can easily gets a constant $A > 0$ such that \begin{equation} | v_m'(t) |^{2} + \| v_m(t) \|^{2} + | \Delta v_m(t) |^{2} + \int_{0}^{t} \| v_{m}'(s) \|^{2} \, ds \leq \, A\, .\label{3.39} \end{equation} Hence for all $T > 0$ we have $(v_{m})_{m \in {\mathbb{N}}}$ bounded in $L{^{\infty}}(0,T;H_{0}^{2}(\Omega))$ and $(v_{m}')_{m \in {\mathbb{N}}}$ bounded in $L{^{\infty}}(0,T;L^{2}(\Omega)) \cap L{^{2}}(0,T;H_{0}^{1}(\Omega))$. From standard compactness arguments we are able to get the existence of global solutions. To complete the proof of Theorem \ref{thm2.1}, we must to establish a rate decay estimate to the total energy of the problem \eqref{1.1}-\eqref{1.3}. In fact, from (\ref{3.33}) and (\ref{3.37}), we get \begin{equation*} \frac{dH}{dt}(t)+ \frac{1}{2} \frac{\| v_m'(t) \|^{2}}{k^{2}(t)} + \frac{1}{2} \frac{\| v_m(t) \|^{2}}{k^{2}(t)} + \frac{1}{4} \frac{| \Delta v_m(t) |^{2}}{k^{4}(t)}\leq 0\,. \end{equation*} From this inequality, (\ref{2.1}) and \eqref{2.3}, we obtain \begin{equation*} \frac{dH}{dt}(t)+ \frac{1}{2C} \frac{| v_m'(t) |^{2}}{k^2_1} + \frac{1}{2} \frac{\| v_m(t) \|^{2}}{k^{2}_1} + \frac{1}{4} \frac{| \Delta v_m(t) |^{2}}{k^{4}_1}\leq 0\,. \end{equation*} From this inequality there exists a positive real constant $\kappa_0$ such that \begin{equation*} \frac{dH}{dt}(t)+ \kappa_0\Big(\frac{3}{4}| v_m'(t) |^{2} + \frac{(1+C^2k_1^2)}{k_0^2}\| v_m(t) \|^{2} + \frac{1}{2k_0^4} | \Delta v_m(t) |^{2}\Big)\leq 0\,, \end{equation*} where \begin{equation} \kappa_0=\min\big\{\frac{2}{3Ck_1^2},\,\frac{k_0^2}{2k_1^2(1+C^2k_1^2)},\, \frac{k_0^4}{2k_1^4}\big\}. \label{3.40} \end{equation} Therefore, by using (\ref{3.35})$_2$ in this inequality we get \begin{equation*} \frac{dH}{dt}(t)+ \frac{1}{\kappa_0}H(t)\leq 0\quad\text{for all } t\geq 0, \end{equation*} which gives \begin{equation} H(t)\leq H(0)\mathrm{e}^{-t/\kappa_0}\quad\text{for all }t\geq 0. \label{3.41} \end{equation} The total energy of the approximate system (\ref{3.11})-(\ref{3.12}) comes from identity (\ref{3.13}); that is, \begin{equation} E(v_m,t) = \frac{1}{2}\big\{ | v_m'(t) |^{2} + \frac{\| v_m(t) \|^{2}}{k^{2}(t)} + \frac{| \Delta v_m(t) |^{2}}{k^{4}(t)}\big\}. \label{3.42} \end{equation} From (\ref{3.42}) and (\ref{3.35})$_1$ there exists $0<\kappa_1\leq 1/2$ such that $\kappa_1 E_m(t)\leq H(t)$. Also we have that $H(0) \leq \alpha (u_{0},u_{1})$. Therefore, from (\ref{3.41}) we get \begin{equation} E(v_m,t)\leq \frac{\alpha (u_{0},u_{1})}{\kappa_1} \mathrm{e}^{-t/\kappa_0}\quad\text{for all}\quad t\geq 0. \label{3.43} \end{equation} The estimate (\ref{3.39}) gives enough convergence to take to the limit $ m \to \infty$ in $E_m$, via Banach-Steinhauss theorem, which implies \begin{equation} E(v,t)\leq \frac{\alpha (u_{0},u_{1})}{\kappa_1}\mathrm{e}^{-t/\kappa_0} \quad\text{for all } t\geq 0, \label{3.44} \end{equation} where \begin{equation} E(v,t) = \frac{1}{2} \big\{| v'(t) |^{2} + \frac{\| v(t) \|^{2}}{k^{2}(t)} + \frac{| \Delta v(t) |^{2}}{k^{4}(t)}\big\}, \label{3.45} \end{equation} is the total energy associated with the system (\ref{3.7})-(\ref{3.9}). Finally, we have to compare the terms of $E(u,t)$ with those of $E(v,t)$. In fact, from (\ref{3.2})-(\ref{3.4}) we have the identities \begin{equation} \begin{gathered} \frac{\partial u}{\partial x_{i}} = \frac{1}{k(t)} \frac{\partial v}{\partial y_{i}} \quad \text{for } i = 1, \dots , n; \\ \frac{\partial u}{\partial t} = -\frac{k'(t)}{k(t)} \sum_{j=1}^{n} \frac{\partial v}{\partial y_{j}}y_{j} + \frac{\partial v}{\partial t}, \quad \Delta u = \frac{1}{k^{2}(t)} \Delta v. \end{gathered}\label{3.46} \end{equation} From the first identity above, we have $$ \|\nabla u(x,t)\|^2_{\mathbb{R}^n} =\frac{1}{k^{2}(t)}\|\nabla v(y,t)\|^2_{\mathbb{R}^n}. $$ Integrating this over $\Omega _t$, using $x=k(t)y$ and $dx=k^n(t)dy$ we get, thanks to hypothesis \eqref{2.3}, that \begin{equation} \begin{gathered} |\nabla u(t)|^2_{L^2(\Omega _t)} \leq \frac{k_1^n}{k^{2}_0}|\nabla v(t)|^2_{L^2(\Omega )}. \label{3.47} \end{gathered} \end{equation} From the second identity of (\ref{3.46}), we obtain $$ \big|\frac{\partial u}{\partial t}(x,t)\big|_\mathbb{R} \leq \frac{k_2}{k_0}\|\nabla v(y,t)\|_{\mathbb{R}^n} \|y\|^2_{\mathbb{R}^n} +\big|\frac{\partial v}{\partial t}(y,t)\big|_\mathbb{R}. $$ Squaring both sides, yields $$ \big|\frac{\partial u}{\partial t}(x,t)\big|_\mathbb{R}^2 \leq \frac{2k_2^2C^2}{k_0^2}\|\nabla v(y,t)\|^2_{\mathbb{R}^n} +2\big|\frac{\partial v}{\partial t}(y,t)\big|^2_\mathbb{R}. $$ Integrating this over $\Omega _t$ and observing that $dx=k^n(t)dy$, we get \begin{equation} \left|u'(t)\right|^2_{L^2(\Omega _t)}\leq \frac{2k_2^2C^2k_1^n}{k_0^2}\left|\nabla v(t)\right|_{L^2(\Omega )}^2+2k_1^n\left| v'(t)\right|^2_{L^2(\Omega )}. \label{3.48} \end{equation} Repeating the same arguments as above in the third identity of (\ref{3.46}), we find \begin{equation} |\Delta u(t)|^2_{L^2(\Omega _t)}\leq \frac{k_1^n}{k_0^4} |\Delta v(t)|_{L^2(\Omega )}^2. \label{3.49} \end{equation} Now, to compare the function $E(u, t)$ with the function $E(v,t)$ it will be used the equivalences of the norms: $\|z\|$ and $|\nabla z|$ in $H^1_0(\Omega )$. Thus, from (\ref{3.47})-(\ref{3.49}) we get \begin{equation} \begin{aligned} E(u, t)&=\frac{1}{2}\big\{|u'(t)|^2_{L^2(\Omega _t)} +|\nabla u(t)|^2_{L^2(\Omega _t)}+|\Delta u(t)|^2_{L^2(\Omega _t)} \big\}\\ &\leq 2k_1^n | v'(t)|^2_{L^2(\Omega )} +\big(\frac{k_1^n}{k^{2}_0}+\frac{2k_2^2C^2k_1^n}{k_0^2}\big) |\nabla v(t)|^2_{L^2(\Omega )}+ \frac{k_1^n}{k_0^4}|\Delta v(t)|_{L^2(\Omega )}^2. \end{aligned} \label{3.50} \end{equation} Thus, choosing \begin{equation} \kappa_2=\max\big\{4k_1^n,\, 2k_1\big(\frac{k_1^n}{k^{2}_0}+\frac{2k_2^2C^2k_1^n}{k_0^2}\big),\, \frac{2k_1^n}{k_0^4}\big\}, \label{3.51} \end{equation} we get from (\ref{3.45}) and (\ref{3.50}) that $E(u,t)\leq \kappa_2 E(v,t)$ for all $t\geq 0$. Therefore, from (\ref{3.44}) we obtain the desired estimate (\ref{2.12}) and consequently the proof of Theorem \ref{thm2.1} is finished \section{Proof of Theorem \ref{thm2.2}} Problems \eqref{1.1}-\eqref{1.3} and (\ref{3.7})-(\ref{3.9}) are equivalent, then it is sufficient to show the uniqueness of solutions to (\ref{3.7})-(\ref{3.9}). Suppose $v$ and $\widehat v$ two solutions of (\ref{3.7})-(\ref{3.9}). Thus, $\phi=v-\widehat v$ satisfies \begin{equation} \begin{aligned} &\phi_{tt}(y,t)- \frac{1}{k^{2}(t)}\Delta ( \phi(y,t)+\phi_t(y,t)) + \frac{1}{k^{4}(t)} \Delta^{2} \phi(y,t) + 2\frac{k'(t)}{k^{3}(t)} \Delta \phi(y,t) \\ & +\frac{k'(t)}{k^{3}(t)} \sum_{j=1}^{n} \Delta (\frac{\partial \phi}{\partial y_{j}}(y,t)) \,y_{j} -2 \frac{k'(t)}{k(t)} \sum_{j=1}^{n} \frac{\partial^{2}\phi}{\partial t \partial y_{j}}(y,t) y_{j}\\ & + \big(\frac{k'(t)}{k(t)}\big)^{2} \sum_{j,\,l=1}^{n} \frac{\partial^{2} \phi}{\partial y_{l} \partial y_{j}}(y,t)y_{l}y_{j} + \big[ \frac{2 (k'(t))^{2} - k(t) k''(t)}{k^{2}(t)} \big] \sum_{j=1}^{n} \frac{\partial\phi}{\partial y_{j}}(y,t) y_{j}\\ &= -\frac{1}{k^2(t)}(\Delta v^2(y,t)-\Delta \widehat v^2(y,t)) \quad \text{in } Q_{\infty}, \end{aligned} \label{4.1} \end{equation} \begin{gather} \phi = \frac{\partial \phi}{\partial \nu} = 0 \quad \text{on}\quad \Sigma_{\infty}, \label{4.2}\\ \phi(y,0)= \phi_t(y,0)=0\quad\text{for}\quad y \in \Omega. \label{4.3} \end{gather} Equation (\ref{4.1}) is given in the sense of $L^2(0,T;H^{-2}(\Omega ))$ and $\phi_t \in L^2(0,T;H^1_0(\Omega ))$, then the duality $\big<\phi_{tt}(t),\phi_t(t)\big>_{H^{-2}(\Omega )\times H^1_0(\Omega )} $ does not make sense. To overcome this difficulty, the uniqueness will be obtained following the argument contained in Ladyzhenskaya-Visik \cite{L-Visik}. In fact, for each $s \in (0,T)$ let $\psi(t)$, be a real function defined for all $t$, in $]0, T[$, by \begin{equation} \psi(y,t)=\begin{cases} -\int_t^s \phi(y,r)dr &\text{if } 0_{H^{-2}(\Omega )\times H^2_0(\Omega )}$ makes sense. Moreover, \begin{equation} \psi_t(y,t)=\phi(y,t)\quad\text{and}\quad \psi(y,s)=0. \label{4.5} \end{equation} Setting $ \psi_1(y,t)=\int_0^t\phi(y,r)dr $, we have \begin{equation} \psi(y,t)=\psi_1(y,t)-\psi_1(y,s)\quad\text{and}\quad \psi(y,0)=-\psi_1(y,s). \label{4.6} \end{equation} Taking the scalar product on $L^2(\Omega)$ of $\psi$ with both sides of (\ref{4.1}) and integrating from $0$, to $s$, yields \begin{align} & \int_0^{s}\langle \phi_{tt}(t), \psi(t)\rangle dt -\int_0^{s}\frac{1}{k^{2}(t)}\langle \Delta ( \phi(t)+\phi_t(t)), \psi(t)\rangle dt + \int_0^{s}\frac{1}{k^4(t)}\langle \Delta^{2} \phi(t), \psi(t)\rangle dt \nonumber\\ &+ 2\int_0^{s}\frac{k'(t)}{k^{3}(t)}\langle \Delta \phi(t), \psi(t)\rangle dt + \int_0^{s}\frac{k'(t)}{k^{3}(t)} \big\langle \sum_{j=1}^{n} \Delta (\frac{\partial \phi}{\partial y_{j}}(t)) \,y_{j}, \psi(t)\big\rangle dt \nonumber\\ &-2\int_0^{s}\frac{k'(t)}{k(t)} \big\langle \sum_{j=1}^{n} \frac{\partial^{2}\phi}{\partial t \partial y_{j}}(t) y_{j}, \psi(t)\big\rangle dt + \int_0^{s}\big(\frac{k'(t)}{k(t)}\big)^{2}\big\langle \sum_{j,\,l=1}^{n} \frac{\partial^{2} \phi}{\partial y_{l} \partial y_{j}}(t)y_{l}y_{j}, \psi(t)\big\rangle dt \nonumber\\ &+\int_0^{s}\big[ \frac{2 (k'(t))^{2} - k(t) k''(t)}{k^{2}(t)} \big] \big\langle \sum_{j=1}^{n} \frac{\partial\phi}{\partial y_{j}}(t) y_{j}, \psi(t)\big\rangle dt \label{4.7}\\ & = -\int_0^{s}\frac{1}{k^2(t)}\langle (\Delta v^2(t) -\Delta {\widehat v}^{\,2}(t)), \psi(t)\rangle dt. \nonumber \end{align} Now, we modify each terms of (\ref{4.7}) by using several times integration by parts, the Green formula, the identities (\ref{4.5}), (\ref{4.6}), the null initial condition (\ref{4.3}), the hypotheses \eqref{2.2}, \eqref{2.3}, (\ref{2.12}) and usual inequalities like: Cauchy-Schwartz, Young so on. In fact, The first term can be changed as \begin{equation} \int_0^{s}\langle \phi_{tt}(t), \psi(t)\rangle dt = (\phi_t(t),\psi(t))\Big|_0^s- \int_0^{s}(\phi_t(t), \phi(t))dt = -\frac{1}{2}|\phi(s)|^2. \label{4.8} \end{equation} The second term of (\ref{4.7}) is modified as \begin{align*} -\int_0^{s}\frac{1}{k^{2}(t)}\langle \Delta \phi(t), \psi(t)\rangle dt &= \int_0^{s}\frac{1}{k^{2}(t)}( \nabla \psi_t(t), \nabla \psi(t)) dt\\ &=\frac 12\int_0^{s}\big\{\frac{d}{dt}\big[\frac{1}{k^{2}(t)} |\nabla \psi(t)|^2\big] - 2\frac{k'(t)}{k^3(t)}|\nabla \psi(t)|^2 \big\}dt \\ &= -\frac 12\frac{1}{k^{2}(0)}|\nabla \psi_1(s)|^2 - \int_0^{s}\frac{k'(t)}{k^3(t)}|\nabla \psi_1(s)|^2 dt. \end{align*} The last integral above is bounded from above as follows: \begin{align*} \big|- \int_0^{s}\frac{k'(t)}{k^3(t)}|\nabla \psi_1(s)|^2 dt\big|_\mathbb{R} &\leq 2\int_0^{s}\frac{|k'(t)|}{k^{3}(t)}|\nabla \psi_1(t)|^2 dt +\frac{2}{k^3_0}|\nabla \psi_1(s)|^2\int_0^{s}|k'(t)|dt \\ &= 2\int_0^{s}\frac{|k'(t)|}{k^{3}(t)}|\nabla \psi_1(t)|^2 dt +\frac{2}{k^3_0}|\nabla \psi_1(s)|^2|k'|_{L^1(0,\infty)}. \end{align*} Therefore, as $k(0)=1$, see \eqref{2.2}, we obtain \begin{equation} \begin{aligned} &-\int_0^{s}\frac{1}{k^{2}(t)}\langle \Delta \phi(t), \psi(t)\rangle dt\\ &\geq -\frac 12|\nabla \psi_1(s)|^2 - 2\int_0^{s}\frac{|k'(t)|}{k^{3}(t)}|\nabla \psi_1(t)|^2 dt -\frac{2}{k_0^3}|k'|_{L^1(0,\infty)}|\nabla \psi_1(s)|^2 \end{aligned}\label{4.9} \end{equation} The third term of (\ref{4.7}) is modified as \begin{align*} -\int_0^{s}\frac{1}{k^{2}(t)}\langle \Delta \phi_t(t), \psi(t)\rangle dt &= -\int_0^{s}( \nabla \phi(t), \big[\frac{1}{k^{2}(t)}\nabla\psi(t)\big]') \\ &= -\int_0^{s}\frac{1}{k^2(t)}|\nabla\phi(t)|^2dt -2\int_0^{s}\frac{k'(t)}{k^3(t)}(\phi(t),\Delta \psi(t))dt. \end{align*} The last integral above is the same that the fifth term of (\ref{4.7}) with positive sign. Thus, we have \begin{equation} -\int_0^{s}\frac{1}{k^{2}(t)}\langle \Delta \phi_t(t), \psi(t)\rangle dt + 2\int_0^{s}\frac{k'(t)}{k^{3}(t)}\langle \Delta \phi(t), \psi(t)\rangle dt =-\int_0^{s}\frac{1}{k^2(t)}|\nabla\phi(t)|^2dt. \label{4.10} \end{equation} Now, we estimate the fourth term of (\ref{4.7}): \begin{equation} \begin{aligned} \int_0^{s}\frac{1}{k^4(t)}\langle \Delta^{2} \phi(t), \psi(t)\rangle dt &=-\frac{1}{2}\left|\Delta \psi_{1}(s)\right|^2 + 2\int_0^{s}\frac{k'(t)}{k^5(t)}|\Delta \psi(t)|^2dt\\ &\geq -\frac{1}{2}\left|\Delta \psi_{1}(s)\right|^2 - \int_0^{s}\frac{|k'(t)|}{k^5_0}|\Delta \psi_1(t)|^2dt \\ &\quad - \frac{1}{k^5_0}|\Delta \psi_1(s)|^2|k'|_{L^1(0,\infty)}. \end{aligned} \label{4.11} \end{equation} The sixth term of (\ref{4.7}) is estimated as \begin{equation} \begin{aligned} &\big|\int_0^{s}\frac{k'(t)}{k^{3}(t)} \langle \sum_{j=1}^{n} \Delta (\frac{\partial \phi}{\partial y_{j}}(t)) \,y_{j}, \psi(t)\rangle dt\big|_\mathbb{R}\\ &=\big|\int_0^{s}\frac{k'(t)}{k^{3}(t)} (\frac{\partial \phi}{\partial y_{j}}(t), 2\frac{\partial \psi}{\partial y_{j}}(t)+ y_j\Delta \psi(t))dt \big|_\mathbb{R} \\ &\leq 2\int_0^{s}\frac{|k'(t)|}{k^{3}(t)}|\nabla\phi(t)| |\nabla \psi_1(s)|dt + \int_0^{s}\frac{|k'(t)|}{k^{3}(t)}|\nabla \phi(t)| \|y\|_{\mathbb{R}^n}|\Delta \psi(t)|dt \\ &\leq \int_0^{s}\big[\frac{3\epsilon_1}{2k^{2}(t)}|\nabla\phi(t)|^2 +\frac{k_2^2}{\epsilon_1 k^{4}_0}|\nabla \psi_1(t)|^2\big]dt +\frac{k_2}{\epsilon_1 k^{4}_0}|\nabla \psi_1(s)|^2|k'|_{L^1(0,\infty)} \\ &\quad +\frac{k_2^2C^2}{2\epsilon_1 k^{4}_0}\int_0^{s} |\Delta \psi_1(t)|^2dt+ \frac{C^2k_2}{2\epsilon_1 k^{4}_0} |\Delta \psi_1(s)|^2|k'|_{L^1(0,\infty)}, \end{aligned}\label{4.12} \end{equation} where $k_2$ is a constant that comes from the hypothesis \eqref{2.2}. That is, $|k'(t)|\leq k_2$. The seventh term of (\ref{4.7}) is estimated as \begin{equation} \begin{aligned} &\big|-2\int_0^{s}\frac{k'(t)}{k(t)} \big\langle \sum_{j=1}^{n} \frac{\partial^{2}\phi}{\partial t \partial y_{j}}(t) y_{j}, \psi(t)\big\rangle dt\big|_\mathbb{R}\\ &= \big|2\int_0^{s}\frac{k'(t)}{k(t)} \Big( \sum_{j=1}^{n} \frac{\partial\phi}{\partial y_{j}}(t) y_{j}, \phi(t)\Big) dt\big|_\mathbb{R} \\ &=\big|\int_0^{s}\frac{k'(t)}{k(t)} (\sum_{j=1}^{n} \frac{\partial}{ \partial y_{j}}[\phi(t)]^2, y_{j}) dt\big|_\mathbb{R}\\ &= \big|\int_0^{s}\frac{n k'(t)}{k(t)} |\phi(t)|^2 dt\big|_\mathbb{R} \\ &\leq \frac{n k_2}{k_0}\int_0^{s} |\phi(t)|^2 dt. \end{aligned} \label{4.13} \end{equation} The eighth term of (\ref{4.7}) is estimated as \begin{equation} \begin{aligned} &\Big|\int_0^{s}\big(\frac{k'(t)}{k(t)}\big)^{2} \big\langle \sum_{j,\,l=1}^{n} \frac{\partial^{2} \phi}{\partial y_{l} \partial y_{j}}(t)y_{l}y_{j}, \psi(t)\big\rangle dt\Big|_\mathbb{R}\\ & = \Big|-\int_0^{s}\big(\frac{k'(t)}{k(t)}\big)^{2}\sum_{j,\,l=1}^{n} \Big(\frac{\partial \phi}{\partial y_{l} }(t), [\delta_{l\,j}y_{j}+y_l]\psi(t)+y_{l}y_{j} \frac{\partial \psi}{\partial y_{j} }(t)\Big) dt\Big|_\mathbb{R}\\ & \leq \int_0^{s}\big(\frac{k'(t)}{k(t)}\big)^{2}\sum_{j,\,l=1}^{n} \big[2\big|\frac{\partial \phi}{\partial y_{l} }(t)\big|d(\Omega ) |\psi(t)|+ \big|\frac{\partial \phi}{\partial y_{l} }(t)\big| [d(\Omega )]^2\big|\frac{\partial \psi}{\partial y_{j} }(t)\big|\big]dt \\ &\leq \int_0^{s}\big(\frac{k'(t)}{k(t)}\big)^{2} \big[2n\sqrt{n} C|\nabla\phi(t)||\psi(t)|+nC^2|\nabla\phi(t)| |\nabla\psi(t)|\big]dt \\ &\leq \epsilon_2\int_0^{s}\frac{1}{k^2(t)}|\nabla\phi(t)|^2dt+ \frac{2n^3C^2k_2^4}{\epsilon_2}\int_0^{s}\frac{1}{k^2(t)} |\psi_1(t)|^2dt \\ &\quad + \frac{2n^2C^2k_2^3}{\epsilon_2k_0^2}|\psi_1(s)|^2|k'|_{L^1(0,\infty)} +\frac{n^2C^4k_2^4}{\epsilon_2}\int_0^{s}\frac{1}{k^2(t)} |\nabla\psi_1(t)|^2dt \\ &\quad + \frac{n^2C^4k_2^3}{2\epsilon_2k_0^2}|\nabla \psi_1(s)|^2|k'|_{L^1(0,\infty)}. \label{4.14} \end{aligned} \end{equation} The ninth term of \eqref{4.7} is estimated as \begin{align} &\Big|\int_0^{s}[ \frac{2 (k'(t))^{2} - k(t) k''(t)}{k^{2}(t)}] \big\langle \sum_{j=1}^{n} \frac{\partial\phi}{\partial y_{j}}(t) y_{j}, \psi(t)\big\rangle dt\Big|_\mathbb{R} \nonumber\\ &=\Big|\int_0^{s}[ -\frac{2 (k'(t))^{2} - k(t) k''(t)}{k^{2}(t)}] \sum_{j=1}^{n}\Big(\phi(t), \psi(t) +y_{j}\frac{\partial\psi}{\partial y_{j}} (t)\Big) dt\Big|_\mathbb{R} \nonumber\\ &\leq \int_0^{s}\big[ \frac{(2k_2+k_1)}{k^{2}(t)}(|k'(t)|+|k''(t)|) \big] [n|\phi(t)||\psi(t)|+d(\Omega )|\phi(t)||\nabla\psi(t)|] dt \nonumber \\ &\leq \big[\frac{n^2(2k_2+k_1)^2}{k^4_0}+\frac{C^2(2k_2+k_1)^2}{4k^4_0} \big]\int_0^{s}|\phi(t)|^2dt \label{4.15} \\ &\quad + 2(C^2+1)\int_0^{s}(|k'(t)|+|k''(t)|)^2|\nabla\psi_1(t)|^2dt \nonumber\\ &\quad + 4(C^2+1)(k_2+k_3)|\nabla\psi_1(s)|^2(|k'|_{L^1(0,\infty)} +|k''|_{L^1(0,\infty)}). \nonumber \end{align} where $k_3$ is a constant due to \eqref{2.2} defined by $|k''(t)|\leq k_3$. The last term of (\ref{4.7}) is estimated as \begin{align} &\Big|-\int_0^{s}\frac{1}{k^2(t)}\langle (\Delta v^2(t) -\Delta {\widehat v}^{\,2}(t)), \psi(t)\rangle dt\Big|_\mathbb{R} \nonumber\\ &=\Big|-\int_0^{s}\frac{1}{k^2(t)}([ v(t)+\widehat v(t)]\phi(t), \Delta\psi(t)) dt\Big|_\mathbb{R} \nonumber\\ &\leq C^2\Big(\|v\|_{L^\infty(0,T; H^1_0(\Omega ))} +\|\widehat v\|_{L^\infty(0,T; H^1_0(\Omega ))}\Big) \int_0^{s}\frac{1}{k^2(t)}|\nabla\phi(t)| |\Delta\psi(t)| dt \nonumber\\ & \leq \epsilon_3\int_0^{s}\frac{1}{k^2(t)}|\nabla\phi(t)|^2dt \nonumber \\ &\quad +\frac{C^4\Big(\|v\|_{L^\infty(0,T; H^1_0(\Omega ))} +\|\widehat v\|_{L^\infty(0,T; H^1_0(\Omega ))}\Big)^2}{2\epsilon_3k_0^2} \int_0^{s}|\Delta\psi_1(t)|^2dt \nonumber\\ &\quad + \frac{C^4\Big(\|v\|_{L^\infty(0,T; H^1_0(\Omega ))} +\|\widehat v\|_{L^\infty(0,T; H^1_0(\Omega ))}\Big)^2s} {2\epsilon_3k_0^2} |\Delta\psi_1(s)|^2. \label{4.16} \end{align} Inserting (\ref{4.8})-(\ref{4.16}) in (\ref{4.7}), we have \begin{equation} \begin{aligned} &\frac{1}{2}|\phi(s)|^2+ \big[\frac{1}{2k_1^2}-K_1[|k'|_{L^1(0,\infty)}+|k''|_{L^1(0,\infty)} ]\big]|\nabla \psi_1(s)|^2 \\ & + [\frac{1}{2k_1^4}-K_2|k'|_{L^1(0,\infty)}-K_3 s] |\Delta \psi_{1}(s)|^2\\ &+[1-(2\epsilon_1+2\epsilon_2+\epsilon_3)] \int_0^{s}\frac{1}{k^2(t)}|\nabla\phi(t)|^2dt \\ &\leq K_4\int_0^{s}[|\phi(t)|^2+|\nabla \psi_1(t)|^2 +|\Delta \psi_1(t)|^2]dt, \end{aligned} \label{4.17} \end{equation} where \begin{equation} \begin{gathered} K_1=\frac{2}{k_0^3}+\frac{k_2}{\epsilon_1 k_0^4} +\frac{2n^2C^3k_2^3}{\epsilon_2k_0^2} +\frac{n^2C^4k_2^3}{2\epsilon_2k_0^2}+4(C^2+1)(k_2+k_3); \\ K_2=\frac{1}{k_0^5}+\frac{C^2k_2}{2\epsilon_1k_0^4}; \\ K_3=\frac{C^4}{2\epsilon_3 k_0^2} \Big(\|v\|_{L^\infty(0,T;H^1_0(\Omega ))} +\|\widehat v\|_{L^\infty(0,T;H^1_0(\Omega ))}\Big)^2; \\ \begin{aligned} K_4&=\frac{4k_2}{k_0^3}+\frac{k_2}{ k_0^5} +\frac{3\epsilon_1}{2 k_0^2}+\frac{k_2^2}{ \epsilon_1 k_0^4} +\frac{k_2^2C^2}{ 2\epsilon_1 k_0^4}+\frac{n k_2}{ k_0} +\frac{\epsilon_2}{ k_0^2}+\frac{2n^3C^3k_2^4}{\epsilon_2} +\frac{n^2C^4k_2^4}{\epsilon_2k_0^2}\\ &\quad +2(C^2+1) +\frac{n^2(2k_2+k_1)^2}{k_0^4} +\frac{C^2(2k_2+k_1)^2}{4k_0^4}+\frac{\epsilon_3}{k_0^2}+K_3. \end{aligned} \end{gathered}\label{4.18} \end{equation} Thus, if $\epsilon_1,\epsilon_2,\epsilon_3$, are taking such that $2\epsilon_1+2\epsilon_2+\epsilon_3\leq 1/2$, and $k',k''$, satisfying the hypothesis (\ref{2.13}), we get \begin{align*} &\frac{1}{2}|\phi(s)|^2 +\frac{1}{4}|\nabla \psi_{1}(s)|^2 +(\frac{1}{4}-K_3 s)|\Delta \psi_{1}(s)|^2 +\frac{1}{2}\int_0^{s}\frac{1}{k^2(t)}|\nabla \phi(t)|^2dt\\ &\leq K_4\int_0^{s}\Big[|\phi(s)|^2 +|\nabla \psi_{1}(s)|^2 +|\Delta \psi_{1}(s)|^2\Big]dt. \end{align*} Now, if $s\leq T_0=1/8K_3$, then \[ |\phi(s)|^2 +|\nabla \psi_{1}(s)|^2+|\Delta \psi_{1}(s)|^2 \leq 8K_4\int_0^{s}\Big[|\phi(s)|^2 +|\nabla \psi_{1}(s)|^2 +|\Delta \psi_{1}(s)|^2\Big]dt. \] From this and Gronwall's inequality we find $\phi(x, s)=0$ a. e. for all $s \in [0,T_0]$. This gives the uniqueness of solutions over the interval $[0,T_0]$. The task now is to show the uniqueness of the solutions over the interval $[T_0, 2T_0]$. In fact, being the solutions unique on $[0,T_0]$ we obtain from (\ref{4.3}) that $$ \phi(y,T_0)= \phi_t(y,T_0)=0\quad\text{for } y \in \Omega. $$ Now, we consider in (\ref{4.4}) the variable $s \in (T_0,T)$ and take $ \psi_1(y,t)=\int_{T_0}^t\phi(y,r)dr $. Thus, we get \[ \psi(y,t)=\psi_1(y,t)-\psi_1(y,s)\quad\text{and}\quad \psi(y,T_0)=-\psi_1(y,s). \] Repeating the steps (\ref{4.7})-(\ref{4.17}) on the whole interval $(T_0,T)\times \Omega $ we get \begin{align*} &\frac{1}{2}|\phi(s)|^2 +\frac{1}{4}|\nabla \psi_{1}(s)|^2 +\Big(\frac{1}{4}-K_3( s-T_0)\Big)|\Delta \psi_{1}(s)|^2+ \frac{1}{2}\int_{T_0}^{s}\frac{1}{k^2(t)}|\nabla \phi(t)|^2dt\\ &\leq K_4\int_{T_0}^{s}\Big[|\phi(s)|^2 +|\nabla \psi_{1}(s)|^2 +|\Delta \psi_{1}(s)|^2\Big]dt. \end{align*} Now, if $s\leq 2T_0=1/4K_3$ then we will have \[ |\phi(s)|^2 +|\nabla \psi_{1}(s)|^2+|\Delta \psi_{1}(s)|^2 \leq 8K_4\int_0^{s}\Big[|\phi(s)|^2 +|\nabla \psi_{1}(s)|^2 +|\Delta \psi_{1}(s)|^2\Big]dt. \] From this and Gronwall's inequality we have $\phi(x, s)=0$ a. e. for all $s \in [T_0,2T_0]$. This gives the uniqueness of solutions over the interval $[0,2T_0]$. 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