Theorem 3.1
(Global existence)
Let \((u_{0},v_{0})\in (H^{2}(\Omega ) \cap H_{0}^{1}(\Omega ))^{2}\), \((u_{1},v_{1})\in (H_{0}^{1}(\Omega ))^{2}\) and \((f_{0},g_{0})\in (H^{1}(\Omega \times (0,1)\times ( \tau _{1}, \tau _{2} ) ))^{2}\) satisfying the compatibility condition
$$\begin{aligned} \bigl(f_{0}(\cdot,0),g_{0}(\cdot,0) \bigr)=(u_{1},v_{1}). \end{aligned}$$
Assume that (A1)–(A2) hold. Then, problem (2.8)–(2.9) admits a weak solution such that \(u,v\in L^{\infty }(0,\infty;H^{2}(\Omega ) \cap H_{0}^{1}(\Omega ))\), \(u_{t},v_{t}\in L^{\infty }(0,\infty; H_{0}^{1}(\Omega ))\), and \(u_{tt},v_{tt}\in L^{2}(0,\infty,H_{0}^{1}(\Omega ))\).
Throughout this section we assume \(( u_{0},v_{0} ) \in ( H^{2}(\Omega )\cap H_{0}^{1}( \Omega ) ) ^{2}\), \(( u_{1},v_{1} ) \in ( H_{0}^{1}(\Omega ) ) ^{2}\) and \(f_{0},g_{0}\in H^{2}(\Omega;H^{1} ( 0,1 ) )\cap H_{0}^{1}( \Omega;H^{1} ( 0,1 ) )\). We employ the Galerkin method to construct a global solution. Let \(T>0\) be fixed and denote by \(V_{k}\) the space generated by \(\{ w_{1},w_{2},\ldots,w_{k} \} \), where the set \(\{ w_{k},k\in \mathbb{N} \} \) is a basis of \(H^{2}(\Omega )\cap H_{0}^{1}(\Omega )\). Now, we define for \(1\leq j\leq k\) the sequence \(\phi _{j} ( x,\rho ) \) as follows:
$$\begin{aligned} \phi _{j} ( x,0 ) =w_{j}. \end{aligned}$$
(3.1)
Then, we may extend \(\phi _{j} ( x,0 ) \) by \(\phi _{j} ( x,\rho ) \) over \(L^{2} ( \Omega \times [ 0,1 ] ) \) and denote by \(Z_{k}\) the space generated by \(\{ \phi _{1},\phi _{2},\ldots,\phi _{k} \} \). We construct approximate solutions \((u^{k},v^{k},z^{k},y^{k} ) \) \(k=1,2,\ldots \) in the form
$$\begin{aligned} &u^{k} ( t ) =\sum_{j=1}^{k}g_{jk} ( t ) w_{j} ( x ), \qquad z^{k} ( t ) =\sum _{j=1}^{k}c_{jk} ( t ) \phi _{j} ( x,\rho,\varrho ) \end{aligned}$$
(3.2)
$$\begin{aligned} &v^{k} ( t ) =\sum_{j=1}^{k}h_{jk} ( t ) w_{j} ( x ), \qquad y^{k} ( t ) =\sum _{j=1}^{k}\,d_{jk} ( t ) \phi _{j} ( x,\rho,\varrho ), \end{aligned}$$
(3.3)
where \(g_{jk,}h_{jk},c_{jk}\), and \(d_{jk}\), \(j=1,2,\ldots\) , are determined by the following ordinary differential equations:
$$\begin{aligned} \textstyle\begin{cases} \langle u_{tt}^{k},w_{j} \rangle +\mu \langle \nabla u^{k},\nabla w_{j} \rangle + ( \lambda +\mu ) \langle \operatorname{div} u^{k},\operatorname{div} w_{j} \rangle +k_{1} \langle u_{t}^{k}, w_{j} \rangle \\ \quad{}+ \langle \int _{\tau _{1}}^{\tau _{2}} \vert \mu _{1} ( \varrho ) \vert z^{k} ( x,1,\varrho,t ) \,d\varrho, w_{j} \rangle = \langle f_{1} ( u^{k},v^{k} ),w_{j} \rangle,\quad 1\leq j\leq k, \\ \langle v_{tt}^{k},w_{j} \rangle +\mu \langle \nabla v^{k},\nabla w_{j} \rangle + ( \lambda +\mu ) \langle \operatorname{div} v^{k}, \operatorname{div} w_{j} \rangle +k_{2} \langle v_{t}^{k},w_{j} \rangle \\ \quad{}+ \langle \int _{\tau _{1}}^{\tau _{2}} \vert \mu _{2} ( \varrho ) \vert y^{k} ( x,1,\varrho,t ) \,d\varrho, w_{j} \rangle = \langle f_{2} ( u^{k},v^{k} ),w_{j} \rangle, \quad 1\leq j\leq k, \\ \langle \varrho z_{t}^{k}+\frac{\partial }{\partial \rho }z^{k}, \phi _{j} \rangle =0, \quad 1\leq j\leq k, \\ \langle \varrho y_{t}^{k}+\frac{\partial }{\partial \rho }y^{k}, \phi _{j} \rangle =0,\quad 1\leq j\leq k, \end{cases}\displaystyle \end{aligned}$$
(3.4)
and
$$\begin{aligned} \textstyle\begin{cases} ( u^{k} ( 0 ),u_{t,k} ( 0 ) ) = ( u_{0k},u_{1k} ), \\ ( v^{k} ( 0 ),v_{t,k} ( 0 ) ) = ( u_{0k},u_{1k} ), \\ z^{k} ( x,0,\varrho,t ) =u_{t}^{k} ( x,t ), \\ y^{k} ( x,0,\varrho,t ) =v_{t}^{k} ( x,t ). \end{cases}\displaystyle \end{aligned}$$
(3.5)
Suppose that
$$\begin{aligned} w_{j}\in H^{2} ( \Omega ). \end{aligned}$$
(3.6)
We choose \(u_{0}^{k}\), \(v_{0}^{k}\), \(u_{1}^{k}\) and \(v_{1}^{k}\in [ w_{1},\ldots,w_{k} ] \), where
$$\begin{aligned} &u^{k} ( 0 ) =u_{0}^{k}=\sum _{j=1}^{k} ( u_{0},w_{j} ) w_{j}\rightarrow u_{0}\quad \text{in }H^{2}(\Omega )\cap H_{0}^{1}( \Omega )\text{ as }k\rightarrow +\infty, \end{aligned}$$
(3.7)
$$\begin{aligned} &u_{t}^{k} ( 0 ) =u_{1}^{k}= \sum_{j=1}^{k} ( u_{1},w_{j} ) w_{j}\rightarrow u_{1}\quad\text{in } H_{0}^{1}( \Omega ) \text{ as }k\rightarrow +\infty, \end{aligned}$$
(3.8)
$$\begin{aligned} & v^{k} ( 0 ) =v_{0}^{k}=\sum _{j=1}^{k} ( v_{0},w_{j} ) w_{j}\rightarrow v_{0}\quad\text{in }H^{2}(\Omega )\cap H_{0}^{1}( \Omega )\text{ as }k\rightarrow +\infty, \end{aligned}$$
(3.9)
$$\begin{aligned} & v_{t}^{k} ( 0 ) =v_{1}^{k}= \sum_{j=1}^{k} ( v_{1},w_{j} ) w_{j}\rightarrow v_{1}\quad \text{in } H_{0}^{1}( \Omega ) \text{ as }k\rightarrow +\infty, \end{aligned}$$
(3.10)
$$\begin{aligned} &z^{k} ( \rho,0 ) =z_{0}^{k}=\sum _{j=1}^{k} ( f_{0}, \phi _{j} ) \phi _{j}\rightarrow f_{0}\quad\text{in }L^{2} \bigl( \Omega \times (0,1)\times (\tau _{1},\tau _{2}) \bigr) \text{ as }k\rightarrow +\infty, \end{aligned}$$
(3.11)
$$\begin{aligned} & y^{k} ( \rho,0 ) =y_{0}^{k}=\sum _{j=1}^{k} ( g_{0}, \phi _{j} ) \phi _{j}\rightarrow g_{0}\quad\text{in }L^{2} \bigl( \Omega \times (0,1)\times (\tau _{1},\tau _{2}) \bigr) \text{ as } k\rightarrow +\infty. \end{aligned}$$
(3.12)
By virtue of the theory of ordinary differential equations, system (3.4)–(3.12) has a unique local solution which is extended to a maximal interval \([ 0,T_{k} [ \) (with \(0< T_{k}\leq +\infty \)). We can utilize a standard compactness argument for the limiting procedure.
The first estimate.
Lemma 3.2
There exists a constant \(T>0\) such that the approximate solution satisfies, for all \(k\geq 1\):
$$\begin{aligned} & u^{k},v^{k} \quad\textit{are bounded in } L^{\infty } \bigl( 0,T;H_{0}^{1} ( \Omega ) \bigr), \end{aligned}$$
(3.13)
$$\begin{aligned} & u_{t}^{k},v_{t}^{k}\quad \textit{are bounded in }L^{\infty } \bigl( 0,T;L^{2} (\Omega ) \bigr), \end{aligned}$$
(3.14)
$$\begin{aligned} & z^{k} ( x,\rho,\varrho,t ),y^{k} ( x,\rho, \varrho,t ) \quad\textit{are bounded in } L^{\infty } \bigl( 0,T; L^{2} \bigl(\Omega \times ( 0,1 ) \times ( \tau _{1}, \tau _{2} ) \bigr) \bigr). \end{aligned}$$
(3.15)
Proof
Multiplying the first and second equations of (3.4) by \(( g_{jk}^{\prime } ) \) and \(( h_{jk}^{\prime } ) \) respectively and summing with respect to j, we obtain
$$\begin{aligned} \begin{aligned} &\frac{1}{2}\frac{d}{dt} \bigl[ \bigl\Vert u_{t}^{k} \bigr\Vert ^{2}+ \mu \bigl\Vert \nabla u^{k} \bigr\Vert ^{2}+ ( \lambda + \mu ) \bigl\Vert \operatorname{div} u^{k} \bigr\Vert ^{2}+ \bigl\Vert v_{t}^{k} \bigr\Vert ^{2}\\ &\quad{}+\mu \bigl\Vert \nabla v^{k} \bigr\Vert ^{2}+ ( \lambda +\mu ) \bigl\Vert \operatorname{div} v^{k} \bigr\Vert ^{2} \bigr] \\ &\quad{}+ \int _{\Omega } u_{t}^{k} \int _{\tau _{1}}^{\tau _{2}}\mu _{1} ( \varrho ) z^{k} ( x,1,\varrho,t ) \,d \varrho \,dx+ \int _{\Omega } v_{t}^{k} \int _{\tau _{1}}^{\tau _{2}}\mu _{2} ( \varrho ) y^{k} ( x,1,\varrho,t ) \,d \varrho \,dx \\ &\quad{}+k_{1} \bigl\Vert u_{t}^{k} \bigr\Vert ^{2}+k_{2} \bigl\Vert v_{t}^{k} \bigr\Vert ^{2} = \bigl\langle f_{1} \bigl( u^{k},v^{k} \bigr),u_{t}^{k} \bigr\rangle + \bigl\langle f_{2} \bigl( u^{k},v^{k} \bigr),v_{t}^{k} \bigr\rangle ,\quad 1\leq j\leq k. \end{aligned} \end{aligned}$$
(3.16)
Multiplying (3.4) by \(\vert \mu _{1} ( \varrho ) \vert ( c_{jk} ) \) and \(\vert \mu _{2} ( \varrho ) \vert ( d_{jk} ) \) respectively, iterating over \(\Omega \times ( 0,1 ) \times ( \tau _{1,}\tau _{2} ) \), and summing with respect to j, we obtain
$$\begin{aligned} \begin{aligned} &\frac{1 }{2}\frac{d}{dt} \int _{\Omega } \int _{0}^{1} \int _{\tau _{1}}^{\tau _{2}}\varrho \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\vert z^{k} \bigr\vert ^{2}\,d \varrho \,d\rho \,dx\\ &\qquad{}+\frac{1}{2} \frac{d}{dt} \int _{\Omega } \int _{0}^{1} \int _{\tau _{1}}^{\tau _{2}}\varrho \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \bigl\vert y^{k} \bigr\vert ^{2}\,d \varrho \,d\rho \,dx \\ &\quad=-\frac{1 }{2} \int _{\Omega } \int _{\tau _{1}}^{\tau _{2}}\varrho \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\vert z^{k} ( x,1, \varrho,t ) \bigr\vert ^{2}\,d\varrho \,dx+ \frac{1}{2} \biggl( \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \,d\varrho \biggr) \bigl\Vert u_{t}^{k} \bigr\Vert ^{2} \\ &\qquad{}-\frac{1}{2} \int _{\Omega } \int _{0}^{1} \int _{\tau _{1}}^{\tau _{2}} \varrho \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \bigl\vert y^{k} ( x,1, \varrho,t ) \bigr\vert ^{2}\,d \varrho \,dx+\frac{1}{2} \biggl( \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \,d\varrho \biggr) \bigl\Vert v_{t}^{k} \bigr\Vert ^{2}. \end{aligned} \end{aligned}$$
(3.17)
By summing (3.16)–(3.17), we have
$$\begin{aligned} &\frac{1}{2}\frac{d}{dt} \bigl[ \bigl\Vert u_{t}^{k} \bigr\Vert ^{2}+ \mu \bigl\Vert \nabla u^{k} \bigr\Vert ^{2}+ ( \lambda + \mu ) \Vert \operatorname{div} u_{k} \Vert ^{2}+ \bigl\Vert v_{t}^{k} \bigr\Vert ^{2} +\mu \bigl\Vert \nabla v^{k} \bigr\Vert ^{2} ( \lambda +\mu ) \bigl\Vert \operatorname{div} v^{k} \bigr\Vert ^{2} \bigr] \\ &\qquad{}+\frac{1 }{2}\frac{d}{dt} \biggl[ \int _{\Omega } \int _{0}^{1} \int _{ \tau _{1}}^{\tau _{2}}\varrho \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\vert z^{k} ( x,\rho, \varrho,t ) \bigr\vert ^{2}\,d\varrho \,d\rho \,dx \\ &\qquad{}+ \int _{\Omega } \int _{0}^{1} \int _{\tau _{1}}^{\tau _{2}}\varrho \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \bigl\vert y^{k} ( x,\rho, \varrho,t ) \bigr\vert ^{2}\,d\varrho \,d\rho \,dx \biggr] \\ \begin{aligned} &\qquad{}+k_{1} \bigl\Vert u_{t}^{k} \bigr\Vert ^{2}+k_{2} \bigl\Vert v_{t}^{k} \bigr\Vert ^{2} \\ &\qquad{}+\frac{1}{2} \int _{\Omega } \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\vert z^{k} ( x,1,\varrho,t ) \bigr\vert ^{2}\,d\varrho \,dx-\frac{1 }{2} \biggl( \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \,d\varrho \biggr) \bigl\Vert u_{t}^{k} \bigr\Vert ^{2} \end{aligned} \\ &\qquad{}+\frac{1 }{2} \int _{\Omega } \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \bigl\vert y^{k} (x,1,\varrho,t ) \bigr\vert ^{2}\,d\varrho \,dx- \frac{1}{2} \biggl( \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \,d\varrho \biggr) \bigl\Vert v_{t}^{k} \bigr\Vert ^{2} \\ &\quad= \int _{\Omega } \bigl( u_{t}^{k}f_{1} \bigl( u^{k},v^{k} \bigr) +v_{t}^{k}f_{2} \bigl( u^{k},v^{k} \bigr) \bigr) \,dx \\ &\qquad{}- \int _{\Omega } u_{t}^{k} \int _{\tau _{1}}^{\tau _{2}}\mu _{1} ( \varrho ) z_{k} ( x,1,\varrho,t ) \,d \varrho \,dx- \int _{\Omega } v_{t}^{k} \int _{\tau _{1}}^{\tau _{2}}\mu _{2} ( \varrho ) y_{k} ( x,1,\varrho,t ) \,d \varrho \,dx. \end{aligned}$$
(3.18)
By using Holder and Young’s inequalities, we have
$$\begin{aligned} \begin{aligned} &\frac{1}{2}\frac{d}{dt} \bigl[ \bigl\Vert u_{t}^{k} \bigr\Vert ^{2}+ \mu \bigl\Vert \nabla u^{k} \bigr\Vert ^{2}+ ( \lambda + \mu ) \bigl\Vert \operatorname{div} u^{k} \bigr\Vert ^{2}\\ &\qquad{}+ \bigl\Vert v_{t}^{k} \bigr\Vert ^{2}+\mu \bigl\Vert \nabla v^{k} \bigr\Vert ^{2}+ ( \lambda +\mu ) \bigl\Vert \operatorname{div} v^{k} \bigr\Vert ^{2} \bigr] \\ &\qquad{}+\frac{1}{2}\frac{d}{dt} \int _{0}^{1} \int _{\tau _{1}}^{\tau _{2}} \varrho \bigl[ \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\Vert z^{k} ( x,\rho, \varrho,t ) \bigr\Vert ^{2} \\ &\qquad{}+\varrho \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \bigl\Vert y^{k} ( x,\rho,\varrho,t ) \bigr\Vert ^{2} \bigr]\,d\varrho \,d\rho \\ &\qquad{}+ \biggl[ k_{1}- \biggl( \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \,d\varrho \biggr) \biggr] \bigl\Vert u_{t}^{k} \bigr\Vert ^{2}+ \biggl[ k_{2}- \biggl( \int _{ \tau _{1}}^{\tau _{2}} \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \,d\varrho \biggr) \biggr] \bigl\Vert v_{t}^{k} \bigr\Vert ^{2} \\ &\quad\leq \int _{\Omega } \bigl( u_{t}^{k}f_{1} \bigl( u^{k},v^{k} \bigr) +v_{t}^{k}f_{2} \bigl( u^{k},v^{k} \bigr) \bigr) \,dx. \end{aligned} \end{aligned}$$
(3.19)
Integrating over \((0,t),0< t< T_{k}\), we obtain
$$\begin{aligned} \begin{aligned} & \bigl\Vert u_{t}^{k} ( t ) \bigr\Vert ^{2}+ \mu \bigl\Vert \nabla u^{k} ( t ) \bigr\Vert ^{2}+ ( \lambda +\mu ) \bigl\Vert \operatorname{div} u^{k} ( t ) \bigr\Vert ^{2}+ \bigl\Vert v_{t}^{k} ( t ) \bigr\Vert ^{2}\\ &\qquad{}+ \mu \bigl\Vert \nabla v^{k} ( t ) \bigr\Vert ^{2}+ ( \lambda +\mu ) \bigl\Vert \operatorname{div} v^{k} ( t ) \bigr\Vert ^{2} \\ &\qquad{}+ \int _{0}^{1} \int _{\tau _{1}}^{\tau _{2}}\varrho \bigl[ \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\Vert z^{k} ( x,\rho, \varrho,t ) \bigr\Vert ^{2} +\varrho \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \bigl\Vert y^{k} ( x,\rho,\varrho,t ) \bigr\Vert ^{2} \bigr]\,d\varrho \,d \rho \\ &\qquad{}+2 \biggl[ k_{1}- \biggl( \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \,d\varrho \biggr) \biggr] \int _{0}^{t} \bigl\Vert u_{t}^{k} ( s ) \bigr\Vert ^{2}\,ds\\ &\qquad{}+2 \biggl[ k_{2}- \biggl( \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \,d\varrho \biggr) \biggr] \int _{0}^{t} \bigl\Vert v_{t}^{k} ( s ) \bigr\Vert ^{2}\,ds \\ &\quad\leq C_{0}+c \int _{0}^{t} \int _{\Omega } \bigl[ u_{t}^{k}f_{1} \bigl( u^{k},v^{k} \bigr)+v_{t}^{k}f_{2} \bigl( u^{k},v^{k} \bigr) \bigr]\,dx\,ds, \end{aligned} \end{aligned}$$
(3.20)
where
$$\begin{aligned} C_{0}={}&C \bigl( \bigl\Vert u_{0}^{k} \bigr\Vert _{H^{1} ( \Omega ) }, \bigl\Vert v_{0}^{k} \bigr\Vert _{H^{1} ( \Omega )}, \bigl\Vert u_{1}^{k} \bigr\Vert _{L^{2} ( \Omega ) }, \bigl\Vert v_{1}^{k} \bigr\Vert _{L^{2} ( \Omega ) },\\ &{} \Vert f_{0} \Vert _{L^{2} (\Omega \times ( 0,1 ) \times ( \tau _{1},\tau _{2} ) ) }, \Vert g_{0} \Vert _{L^{2} (\Omega \times ( 0,1 ) \times ( \tau _{1},\tau _{2} ) ) } \bigr) \end{aligned}$$
is a positive constant. We just need to estimate the right-hand terms of (3.20). Applying Hölder’s inequality, Sobolev’s embedding theorem, and Young’s inequality, we obtain
$$\begin{aligned} \begin{aligned} & \biggl\vert \int _{0}^{t} \int _{\Omega }u_{t}^{k} ( s ) f_{1} \bigl( u^{k} ( s ),v^{k} ( s ) \bigr) \,ds \biggr\vert \\ &\quad\leq C \int _{0}^{t} \int _{\Omega } \bigl( \bigl\vert u^{k} ( s ) \bigr\vert ^{p}+ \bigl\vert v^{k} ( s ) \bigr\vert ^{p}+ \bigl\vert u^{k} ( s ) \bigr\vert ^{ \frac{p-1}{2}} \bigl\vert v^{k} ( s ) \bigr\vert ^{ \frac{p+1}{2}} \bigr) \bigl\vert u_{t}^{k} ( s ) \bigr\vert \,ds \\ &\quad\leq C \int _{0}^{t} \int _{\Omega } \bigl( \bigl\Vert u^{k} ( s ) \bigr\Vert _{2p}^{p}+ \bigl\Vert v^{k} ( s ) \bigr\Vert _{2p}^{p}\\ &\qquad{}+ \bigl\Vert u^{k} ( s ) \bigr\Vert _{3 ( p-1 ) }^{\frac{p-1}{2}} \bigl\Vert v^{k} ( s ) \bigr\Vert _{\frac{3 ( p+1 ) }{2}}^{\frac{p+1}{2}} \bigr) \bigl\Vert u_{t}^{k} ( s ) ( s ) \bigr\Vert _{2}\,ds \\ &\quad\leq C \int _{0}^{t} \int _{\Omega } \bigl( \bigl\Vert \nabla u^{k} ( s ) \bigr\Vert _{2}^{p}+ \bigl\Vert \nabla v^{k} ( s ) \bigr\Vert _{2}^{p}\\ &\qquad{}+ \bigl\Vert \nabla u^{k} ( s ) \bigr\Vert _{2}^{\frac{p-1}{2}} \bigl\Vert \nabla v^{k} ( s ) \bigr\Vert _{2}^{\frac{p+1}{2}} \bigr) \bigl\Vert u_{t}^{k} ( s ) ( s ) \bigr\Vert _{2}\,ds \\ &\quad\leq C \int _{0}^{t} \int _{\Omega } \bigl( \bigl\Vert u_{t}^{k} ( s ) \bigr\Vert _{2}^{2}+ \bigl\Vert \nabla u^{k} ( s ) \bigr\Vert _{2}^{2p}+ \bigl\Vert \nabla v^{k} ( s ) \bigr\Vert _{2}^{2p}\\ &\qquad{}+ \bigl\Vert \nabla u^{k} ( s ) \bigr\Vert _{2}^{ ( p-1 ) } \bigl\Vert \nabla v^{k} ( s ) \bigr\Vert _{2}^{ ( p+1 ) } \bigr) \,ds, \end{aligned} \end{aligned}$$
(3.21)
when we have used in (3.21) the Sobolev imbedding in (2.5) and the fact when \(n=3\) then \(2p=3 ( p-1 ) =\frac{3 ( p+1 ) }{2}=6\).
Likewise, we obtain
$$\begin{aligned} \begin{aligned} & \biggl\vert \int _{0}^{t} \int _{\Omega }v_{t}^{k} ( s ) f_{2} \bigl( u^{k} ( s ),v^{k} ( s ) \bigr) \,ds \biggr\vert \\ &\quad\leq C \int _{0}^{t} \int _{\Omega } \bigl( \bigl\Vert v_{t}^{k} ( s ) \bigr\Vert _{2}^{2}+ \bigl\Vert \nabla u^{k} ( s ) \bigr\Vert _{2}^{2p}+ \bigl\Vert \nabla v^{k} ( s ) \bigr\Vert _{2}^{2p}\\ &\qquad{}+ \bigl\Vert \nabla u^{k} ( s ) \bigr\Vert _{2}^{p+1} \bigl\Vert \nabla v^{k} ( s ) \bigr\Vert _{2}^{p-1} \bigr) \,ds. \end{aligned} \end{aligned}$$
(3.22)
Let
$$\begin{aligned} X_{k} ( t ) = \bigl\Vert u_{t}^{k} ( t ) \bigr\Vert _{2}^{2}+ \bigl\Vert v_{t}^{k} ( t ) \bigr\Vert _{2}^{2}+ \bigl\Vert \nabla u^{k} ( t ) \bigr\Vert _{2}^{2}+ \bigl\Vert \nabla v^{k} ( t ) \bigr\Vert _{2}^{2}. \end{aligned}$$
(3.23)
From assumptions of Lemma 2.1, we can find positive constants such that
$$\begin{aligned} \begin{aligned} &X_{k} ( t ) +c_{1} \bigl\Vert \operatorname{div} u^{k} ( t ) \bigr\Vert ^{2}+c_{1} \bigl\Vert \operatorname{div} v^{k} ( t ) \bigr\Vert ^{2} \\ &\qquad{}+c_{2} \int _{0}^{1} \int _{\tau _{1}}^{\tau _{2}}\varrho \bigl[ \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\Vert z^{k} ( x,\rho, \varrho,t ) \bigr\Vert ^{2} + \varrho \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \bigl\Vert y^{k} ( x,\rho, \varrho,t ) \bigr\Vert ^{2} \bigr]\,d\varrho \,d\rho \\ &\qquad{}+c_{3} \int _{0}^{t} \bigl\Vert u_{t}^{k} ( s ) \bigr\Vert ^{2}\,ds+c_{3} \int _{0}^{t} \bigl\Vert v_{t}^{k} ( s ) \bigr\Vert ^{2}\,ds \\ &\quad\leq C_{0}+c \int _{0}^{t} \int _{\Omega } \bigl( X_{k} ( t ) \bigr) ^{p} \,dx\,ds. \end{aligned} \end{aligned}$$
(3.24)
Particulary, we have
$$\begin{aligned} X_{k} ( t ) \leq C_{0}+c \int _{0}^{t} \int _{\Omega } \bigl( X_{k} ( t ) \bigr) ^{p} \,dx\,ds. \end{aligned}$$
(3.25)
Using Gronwall-type inequality, we can get
$$\begin{aligned} X_{k} ( t ) \leq \bigl[ C_{0}- ( p-1 ) Ct \bigr]^{-1/ ( p-1 ) }. \end{aligned}$$
(3.26)
Thus, we deduce from (3.26) that there exists a time \(T>0\) such that
$$\begin{aligned} X_{k} ( t ) \leq C_{1},\quad \forall t\in [ 0,T ], \end{aligned}$$
(3.27)
where \(C_{1}\) is a positive constant independent of k. Then inequality (3.27) established the first two parts of lemma. The last part of lemma immediately follows from (3.24). □
The second estimate: First, we are going to estimate \(u_{tt}^{k} ( 0 ) \) and \(v_{tt}^{k} ( 0 ) \). Testing the first and second equations in (3.4) by \(g_{j,k}^{\prime \prime } ( t ) \) and \(h_{j,k}^{\prime \prime } ( t ) \) respectively and taking \(t=0\), we obtain
$$\begin{aligned} \begin{aligned}& \bigl\Vert u_{tt}^{k} ( 0 ) \bigr\Vert _{2}^{2}+ \bigl\Vert v_{tt}^{k} ( 0 ) \bigr\Vert _{2}^{2} \\ &\quad\leq \bigl\Vert \Delta u_{0}^{k} \bigr\Vert _{2}^{2}+ \bigl\Vert \Delta v_{0}^{k} \bigr\Vert _{2}^{2}+c \bigl\Vert \operatorname{div} u_{0}^{k} \bigr\Vert _{2}+c \bigl\Vert \operatorname{div} v_{0}^{k} \bigr\Vert _{2}^{2}+ \bigl\Vert \Delta u_{t}^{k} ( 0 ) \bigr\Vert _{2}^{2} \\ &\qquad{}+ \bigl\Vert \Delta v_{t}^{k} ( 0 ) \bigr\Vert _{2}^{2}+c \int _{\Omega } \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\vert \Delta z^{k} (x,1, \varrho,0 ) \bigr\vert ^{2} \,d\varrho \,dx \\ &\qquad{}+c \int _{\Omega } \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \bigl\vert \Delta y^{k} ( x,1, \varrho,0 ) \bigr\vert ^{2} \,d\varrho \,dx. \end{aligned} \end{aligned}$$
(3.28)
From (3.7)–(3.12), we have
$$\begin{aligned} \bigl\Vert u_{tt}^{k} ( 0 ) \bigr\Vert _{2}^{2}+ \bigl\Vert v_{tt}^{k} ( 0 ) \bigr\Vert _{2}^{2}\leq C. \end{aligned}$$
(3.29)
In order to calculate the second estimate, we take the derivatives of the first and second equations of system (3.4) with respect to t, we get
$$\begin{aligned} \begin{aligned} & \bigl\langle u_{ttt}^{k},w_{j} \bigr\rangle +\mu \bigl\langle \nabla u_{t}^{k},\nabla w_{j} \bigr\rangle + ( \lambda +\mu ) \bigl\langle \operatorname{div} u_{t}^{k}, \operatorname{div} w_{j} \bigr\rangle \\ &\quad{}+k_{1} \bigl\langle u_{tt}^{k}, w_{j} \bigr\rangle + \biggl\langle \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{1} ( \varrho ) \bigr\vert z_{t}^{k} ( x,1,\varrho,t ) \,d \varrho, w_{j} \biggr\rangle = \bigl\langle Df_{1} ( u_{k},v_{k} ),w_{j} \bigr\rangle ,\quad 1\leq j\leq k \end{aligned} \end{aligned}$$
and
$$\begin{aligned} \begin{aligned} & \bigl\langle v_{ttt}^{k},w_{j} \bigr\rangle +\mu \bigl\langle \nabla v_{t}^{k},\nabla w_{j} \bigr\rangle + ( \lambda +\mu ) \bigl\langle \operatorname{div} v_{t}^{k},\operatorname{div} w_{j} \bigr\rangle \\ &\quad{}+k_{2} \bigl\langle v_{tt}^{k}, w_{j} \bigr\rangle + \biggl\langle \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{2} ( \varrho ) \bigr\vert y_{t}^{k} ( x,1,\varrho,t ) \,d \varrho, w_{j} \biggr\rangle = \bigl\langle Df_{2} ( u_{k},v_{k} ),w_{j} \bigr\rangle ,\quad 1\leq j\leq k. \end{aligned} \end{aligned}$$
Multiplying by \(( g_{j,k}^{\prime \prime } ( t ) ) \) and \(( h_{j,k}^{\prime \prime } ( t ) ) \) respectively and summing with respect to j from 1 to k, we obtain
$$\begin{aligned} \begin{aligned} &\frac{1}{2}\frac{d}{dt} \bigl[ \bigl\Vert u_{tt}^{k} ( t ) \bigr\Vert ^{2}+\mu \bigl\Vert \nabla u_{t}^{k} ( t ) \bigr\Vert ^{2}+ ( \lambda +\mu ) \bigl\Vert \operatorname{div} u_{t}^{k} ( t ) \bigr\Vert ^{2} \bigr] +k_{1} \bigl\Vert u_{tt}^{k} ( t ) \bigr\Vert ^{2} \\ &\qquad{}+ \int _{\Omega } u_{tt}^{k} ( t ) \int _{\tau _{1}}^{ \tau _{2}} \bigl\vert \mu _{1} ( \varrho ) \bigr\vert z_{t}^{k} ( x,1,\varrho,t ) \,d\varrho \,dx\\ &\quad= \int _{\Omega }Df_{1} ( u_{k},v_{k} ) u_{tt}^{k} ( t ) \,dx,\quad 1 \leq j\leq k, \end{aligned} \end{aligned}$$
(3.30)
and
$$\begin{aligned} \begin{aligned} &\frac{1}{2}\frac{d}{dt} \bigl[ \bigl\Vert v_{tt}^{k} \bigr\Vert ^{2}+ \mu \bigl\Vert \nabla v_{t}^{k} \bigr\Vert ^{2}+ ( \lambda +\mu ) \bigl\Vert \operatorname{div} v_{t}^{k} \bigr\Vert ^{2} \bigr] +k_{2} \bigl\Vert v_{tt}^{k} \bigr\Vert ^{2} \\ &\qquad{} + \int _{\Omega } v_{tt}^{k} ( t ) \int _{\tau _{1}}^{ \tau _{2}} \bigl\vert \mu _{2} ( \varrho ) \bigr\vert y_{t}^{k} ( x,1,\varrho,t ) \,d\varrho \,dx\\ &\quad = \int _{\Omega }Df_{2} ( u_{k},v_{k} ) v_{tt}^{k} ( t ) \,dx, \quad 1 \leq j\leq k. \end{aligned} \end{aligned}$$
(3.31)
Differentiating the third and fourth equations in (3.4) with respect to t, we get
$$\begin{aligned} &\biggl\langle \varrho z_{tt}^{k}+\frac{\partial }{\partial \rho } z_{t}^{k}, \phi _{j} \biggr\rangle =0,\quad 1\leq j \leq k, \\ &\biggl\langle \varrho y_{tt}^{k}+\frac{\partial }{\partial \rho } y_{t}^{k}, \phi _{j} \biggr\rangle =0, \quad 1 \leq j\leq k. \end{aligned}$$
Multiplying by \(\vert \mu _{1} ( \varrho ) \vert c_{jk}^{\prime }\) and \(\vert \mu _{2} ( \varrho ) \vert \,d_{jk}^{\prime }\) respectively, integrating over \(( 0,1 ) \times ( \tau _{1},\tau _{2} ) \), and summing over j from 1 to k, it follows that
$$\begin{aligned} \begin{aligned} &\frac{1}{2}\frac{d}{dt} \int _{0}^{1} \int _{\tau _{1}}^{ \tau _{2}} \int _{\Omega }\varrho \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\vert z_{t}^{k} ( x,\rho, \varrho,t ) \bigr\vert ^{2}\,d\varrho \,d\rho \,dx\\ &\quad{}+\frac{1}{2}\int _{0}^{1} \int _{\Omega } \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \frac{d}{d\rho } \bigl\vert z_{t}^{k} ( x,\rho,\varrho,t ) \bigr\vert ^{2}\,d \varrho \,d\rho \,dx=0, \\ & \frac{1}{2}\frac{d}{dt} \int _{0}^{1} \int _{\tau _{1}}^{\tau _{2}} \int _{\Omega }\varrho \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\vert y_{t}^{k} ( x,\rho,\varrho,t ) \bigr\vert ^{2}\,d\varrho \,d\rho \,dx\\ &\quad{}+\frac{1}{2}\int _{0}^{1} \int _{\Omega } \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \frac{d}{d\rho } \bigl\vert y_{t}^{k} ( x,\rho,\varrho,t ) \bigr\vert ^{2}\,d \varrho \,d\rho \,dx=0, \end{aligned} \end{aligned}$$
then we obtain
$$\begin{aligned} \begin{aligned} &\frac{1}{2}\frac{d}{dt} \int _{0}^{1} \int _{\Omega } \int _{\tau _{1}}^{\tau _{2}}\varrho \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\vert z_{t}^{k} ( x,\rho,\varrho,t ) \bigr\vert ^{2}\,d\varrho \,d\rho \,dx\\ &\quad{}+ \frac{1}{2} \int _{\Omega } \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\vert z_{t}^{k} ( x,1,\varrho,t ) \bigr\vert ^{2}\,d\varrho \,dx \\ &\quad{}-\frac{1}{2} \biggl( \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \,d\varrho \biggr) \bigl\Vert u_{tt}^{k} ( t ) \bigr\Vert ^{2}\,dx=0 \end{aligned} \end{aligned}$$
(3.32)
and
$$\begin{aligned} \begin{aligned} &\frac{1}{2}\frac{d}{dt} \int _{0}^{1} \int _{\tau _{1}}^{ \tau _{2}} \int _{\Omega }\varrho \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \bigl\vert y_{t}^{k} ( x,\rho,\varrho,t ) \bigr\vert ^{2}\,d\varrho \,d\rho \,dx\\ &\quad{}+\frac{1}{2} \int _{\Omega } \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \bigl\vert y_{t}^{k} ( x,1,\varrho,t ) \bigr\vert ^{2}\,d\varrho\, dx \\ &\quad{} -\frac{1}{2} \biggl( \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \,d\varrho \biggr) \bigl\Vert v_{tt}^{k} ( t ) \bigr\Vert ^{2}\,dx=0. \end{aligned} \end{aligned}$$
(3.33)
Taking the sum of (3.30), (3.31),(3.32), and (3.33), we get
$$\begin{aligned} &\frac{1}{2}\frac{d}{dt} \bigl[ \bigl\Vert u_{tt}^{k} \bigr\Vert ^{2}+ \mu \bigl\Vert \nabla u_{t}^{k} \bigr\Vert ^{2}+ ( \lambda +\mu ) \bigl\Vert \operatorname{div} u_{t}^{k} \bigr\Vert ^{2}+ \bigl\Vert v_{tt}^{k} \bigr\Vert ^{2}+\mu \bigl\Vert \nabla v_{t}^{k} \bigr\Vert ^{2}+ ( \lambda +\mu ) \bigl\Vert \operatorname{div} v_{t}^{k} \bigr\Vert ^{2} \bigr] \\ &\qquad{} +\frac{1}{2}\frac{d}{dt} \biggl[ \int _{0}^{1} \int _{\Omega } \int _{ \tau _{1}}^{\tau _{2}}\varrho \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\vert z_{t}^{k} ( x,\rho,\varrho,t ) \bigr\vert ^{2}\,d\varrho \,d\rho\, dx \\ &\qquad{}+ \int _{0}^{1} \int _{\tau _{1}}^{ \tau _{2}} \int _{\Omega }\varrho \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \bigl\vert y_{t}^{k} ( x,\rho,\varrho,t ) \bigr\vert ^{2}\,d\varrho \,d\rho\, dx \biggr] \\ &\qquad{} +k_{1} \bigl\Vert u_{tt}^{k} \bigr\Vert ^{2}+k_{2} \bigl\Vert v_{tt}^{k} \bigr\Vert ^{2}+\frac{1}{2} \int _{\Omega } \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\vert z_{t}^{k} ( x,1,\varrho,t ) \bigr\vert ^{2}\,d\varrho\, dx \\ &\qquad{} +\frac{1}{2} \int _{\Omega } \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \bigl\vert y_{t}^{k} ( x,1, \varrho,t ) \bigr\vert ^{2}\,d\varrho\, dx \\ &\quad=\frac{1}{2} \biggl( \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \,d\varrho \biggr) \bigl\Vert u_{tt}^{k} ( t ) \bigr\Vert ^{2}+\frac{1}{2} \biggl( \int _{\tau _{1}}^{ \tau _{2}} \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \,d\varrho \biggr) \bigl\Vert v_{tt}^{k} ( t ) \bigr\Vert ^{2} \\ &\qquad{}- \int _{\Omega } u_{tt}^{k} ( t ) \int _{\tau _{1}}^{ \tau _{2}} \bigl\vert \mu _{1} ( \varrho ) \bigr\vert z_{t}^{k} ( x,1,\varrho,t ) \,d\varrho \, dx \\ &\qquad{}- \int _{\Omega } v_{tt}^{k} ( t ) \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{2} ( \varrho ) \bigr\vert y_{t}^{k} ( x,1,\varrho,t ) \,d\varrho \, dx \\ &\qquad{}+ \int _{\Omega }Df_{1} ( u_{k},v_{k} ) u_{tt}^{k} ( t ) \,dx+ \int _{\Omega }Df_{2} ( u_{k},v_{k} ) v_{tt}^{k} ( t ) \,dx. \end{aligned}$$
(3.34)
Using Cauchy–Schwarz and Young’s inequalities, we conclude
$$\begin{aligned} \begin{aligned}& \biggl\vert \int _{\Omega } u_{tt}^{k} ( t ) \int _{\tau _{1}}^{\tau _{2}} \biggr\vert \mu _{1} ( \varrho ) \bigl\vert z_{t}^{k} ( x,1,\varrho,t ) \,d \varrho\, dx \bigr\vert \\ &\quad\leq \frac{1}{2} \biggl( \int _{\tau _{1}}^{ \tau _{2}} \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \,d \varrho \biggr) \bigl\Vert u_{tt}^{k} ( t ) \bigr\Vert ^{2} \\ &\qquad{}+\frac{1}{2} \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\Vert z_{t}^{k} ( x,1, \varrho,t ) \bigr\Vert ^{2}\,d\varrho. \end{aligned} \end{aligned}$$
(3.35)
Similarly,
$$\begin{aligned} \begin{aligned} &\biggl\vert \int _{\Omega } v_{tt}^{k} ( t ) \int _{\tau _{1}}^{\tau _{2}} \biggr\vert \mu _{2} ( \varrho ) \bigl\vert y_{t}^{k} ( x,1,\varrho,t ) \,d \varrho\, dx \bigr\vert \\ &\quad\leq \frac{1}{2} \biggl( \int _{\tau _{1}}^{ \tau _{2}} \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \,d \varrho \biggr) \bigl\Vert v_{tt}^{k} ( t ) \bigr\Vert ^{2} \\ &\qquad{}+\frac{1}{2} \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \bigl\Vert y_{t}^{k} ( x,1, \varrho,t ) \bigr\Vert ^{2}\,d\varrho. \end{aligned} \end{aligned}$$
(3.36)
For the source term
$$\begin{aligned} &\int _{\Omega }Df_{1} ( u_{k},v_{k} ) u_{tt}^{k} ( t ) \,dx \\ \begin{aligned} &\quad\leq C \bigl[ \bigl( \bigl\Vert u^{k} \bigr\Vert ^{p-1}+ \bigl\Vert v^{k} \bigr\Vert ^{p-1} \bigr) \bigl\Vert u_{t}^{k} \bigr\Vert + \bigl( \bigl\Vert u^{k} \bigr\Vert ^{p-1}+ \bigl\Vert v^{k} \bigr\Vert ^{p-1} \bigr) \bigl\Vert v_{t}^{k} \bigr\Vert \bigr] \bigl\Vert u_{tt}^{k} \bigr\Vert \\ &\quad\leq C \bigl[ \bigl\Vert u^{k} \bigr\Vert _{2} ^{2(p-1)}+ \bigl\Vert v^{k} \bigr\Vert _{2}^{2(p-1)}+ \bigl\Vert u_{t}^{k} \bigr\Vert _{2}^{2} + \bigl\Vert v_{t}^{k} \bigr\Vert _{2}^{2} \bigr] \bigl\Vert u_{tt}^{k} \bigr\Vert _{2} \end{aligned} \\ &\quad\leq \bigl\Vert u_{tt}^{k} ( t ) \bigr\Vert _{2}^{2}+c \end{aligned}$$
(3.37)
and
$$\begin{aligned} \begin{aligned} &\int _{\Omega }Df_{2} ( u_{k},v_{k} ) v_{tt}^{k} ( t ) \,dx\\ &\quad\leq C \bigl[ \bigl( \bigl\Vert u^{k} \bigr\Vert ^{p-1}+ \bigl\Vert v^{k} \bigr\Vert ^{p-1} \bigr) \bigl\Vert u_{t}^{k} \bigr\Vert + \bigl( \bigl\Vert u^{k} \bigr\Vert ^{p-1}+ \bigl\Vert v^{k} \bigr\Vert ^{p-1} \bigr) \bigl\Vert v_{t}^{k} \bigr\Vert \bigr] \bigl\Vert v_{tt}^{k} \bigr\Vert \\ &\quad\leq C \bigl[ \bigl\Vert u^{k} \bigr\Vert _{2} ^{2(p-1)}+ \bigl\Vert v^{k} \bigr\Vert _{2}^{2(p-1)}+ \bigl\Vert u_{t}^{k} \bigr\Vert _{2}^{2} + \bigl\Vert v_{t}^{k} \bigr\Vert _{2}^{2} \bigr] \bigl\Vert v_{tt}^{k} \bigr\Vert _{2} \\ &\quad\leq \bigl\Vert v_{tt}^{k} ( t ) \bigr\Vert _{2}^{2}+c. \end{aligned} \end{aligned}$$
(3.38)
Taking into account (3.35)–(3.38) into (3.34), we get
$$\begin{aligned} \begin{aligned} &\frac{1}{2}\frac{d}{dt} \bigl[ \bigl\Vert u_{tt}^{k} \bigr\Vert ^{2}+\mu \bigl\Vert \nabla u_{t}^{k} \bigr\Vert ^{2}+ ( \lambda +\mu ) \bigl\Vert \operatorname{div} u_{t}^{k} \bigr\Vert ^{2}+ \bigl\Vert v_{tt}^{k} \bigr\Vert ^{2}+\mu \bigl\Vert \nabla v_{t}^{k} \bigr\Vert ^{2}+ ( \lambda +\mu ) \bigl\Vert \operatorname{div} v_{t}^{k} \bigr\Vert ^{2} \bigr] \\ &\qquad{} +\frac{1}{2}\frac{d}{dt} \biggl[ \int _{0}^{1} \int _{\Omega } \int _{ \tau _{1}}^{\tau _{2}}\varrho \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\vert z_{t}^{k} ( x,\varrho,\rho,t ) \bigr\vert ^{2}\,d\varrho \,d\rho\, dx\\ &\qquad{}+ \frac{1}{2} \int _{0}^{1} \int _{ \tau _{1}}^{\tau _{2}} \int _{\Omega }\varrho \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\vert y_{t}^{k} ( x,\varrho, \rho,t ) \bigr\vert ^{2}\,d\varrho \,d\rho\, dx \biggr] \\ &\qquad{}+k_{1} \bigl\Vert u_{tt}^{k} \bigr\Vert ^{2}+k_{2} \bigl\Vert v_{tt}^{k} \bigr\Vert ^{2}+\frac{1}{2} \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\Vert z_{t}^{k} ( x,1,\varrho,t ) \bigr\Vert ^{2}\,d\varrho\\ &\qquad{} +\frac{1}{2} \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \bigl\Vert y_{t}^{k} ( x,1,\varrho,t ) \bigr\Vert ^{2}\,d\varrho \\ &\quad \leq \frac{1}{2} \biggl( \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \,d\varrho \biggr) \bigl\Vert u_{tt}^{k} ( t ) \bigr\Vert ^{2}+\frac{1}{2} \biggl( \int _{\tau _{1}}^{ \tau _{2}} \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \,d\varrho \biggr) \bigl\Vert v_{tt}^{k} ( t ) \bigr\Vert ^{2} \\ &\qquad{} + \biggl( \frac{\int _{\tau _{1}}^{\tau _{2}} \vert \mu _{1} ( \varrho ) \vert \,d\varrho }{2} \biggr) \bigl\Vert u_{tt}^{k} \bigr\Vert ^{2}+ \biggl( \frac{\int _{\tau _{1}}^{\tau _{2}} \vert \mu _{2} ( \varrho ) \vert \,d\varrho }{2} \biggr) \bigl\Vert v_{tt}^{k} \bigr\Vert ^{2} \\ & \qquad{}+\frac{1}{2} \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\Vert z_{t}^{k} ( x,1, \varrho,t ) \bigr\Vert ^{2}\,d\varrho +\frac{1}{2} \int _{\tau _{1}}^{ \tau _{2}} \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \bigl\Vert y_{t}^{k} ( x,1,\varrho,t ) \bigr\Vert ^{2}\,d \varrho \\ &\qquad{}+ \bigl\Vert u_{tt}^{k} ( t ) \bigr\Vert ^{2} + \bigl\Vert v_{tt}^{k} ( t ) \bigr\Vert ^{2}+c. \end{aligned} \end{aligned}$$
After simplification, we obtain
$$\begin{aligned} &\frac{1}{2}\frac{d}{dt} \bigl[ \bigl\Vert u_{tt}^{k} \bigr\Vert ^{2}+ \mu \bigl\Vert \nabla u_{t}^{k} \bigr\Vert ^{2}+ ( \lambda +\mu ) \bigl\Vert \operatorname{div} u_{t}^{k} \bigr\Vert ^{2}+ \bigl\Vert v_{tt}^{k} \bigr\Vert ^{2}+\mu \bigl\Vert \nabla v_{t}^{k} \bigr\Vert ^{2}+ ( \lambda +\mu ) \bigl\Vert \operatorname{div} v_{t}^{k} \bigr\Vert ^{2} \bigr] \\ &\qquad{} +\frac{1}{2}\frac{d}{dt} \biggl[ \int _{0}^{1} \int _{\Omega } \int _{ \tau _{1}}^{\tau _{2}}\varrho \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\vert z_{t}^{k} ( x,\varrho,\rho,t ) \bigr\vert ^{2}\,d\varrho \,d\rho\, dx \\ \begin{aligned} &\qquad{}+ \frac{1}{2} \int _{0}^{1} \int _{ \tau _{1}}^{\tau _{2}} \int _{\Omega }\varrho \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\vert y_{t}^{k} ( x,\varrho, \rho,t ) \bigr\vert ^{2}\,d\varrho \,d\rho\, dx \biggr] \\ &\qquad{}+k_{1} \bigl\Vert u_{tt}^{k} \bigr\Vert ^{2}+k_{2} \bigl\Vert v_{tt}^{k} \bigr\Vert ^{2} \end{aligned} \\ &\quad \leq \biggl( \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \,d\varrho \biggr) \bigl\Vert u_{tt}^{k} ( t ) \bigr\Vert ^{2}+ \biggl( \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \,d\varrho \biggr) \bigl\Vert v_{tt}^{k} ( t ) \bigr\Vert ^{2} \\ &\qquad{}+ \bigl\Vert u_{tt}^{k} ( t ) \bigr\Vert ^{2} + \bigl\Vert v_{tt}^{k} ( t ) \bigr\Vert ^{2}+c. \end{aligned}$$
(3.39)
Integrating (3.39) over \(( 0,t ) \), we get
$$\begin{aligned} & \bigl[ \bigl\Vert u_{tt}^{k} (t ) \bigr\Vert ^{2}+\mu \bigl\Vert \nabla u_{t}^{k} (t ) \bigr\Vert ^{2}+ ( \lambda +\mu ) \bigl\Vert \operatorname{div} u_{t}^{k} (t ) \bigr\Vert ^{2}+ \bigl\Vert v_{tt}^{k} (t ) \bigr\Vert ^{2}\\ &\qquad{}+\mu \bigl\Vert \nabla v_{t}^{k} (t ) \bigr\Vert ^{2}+ ( \lambda +\mu ) \bigl\Vert \operatorname{div} v_{t}^{k} (t ) \bigr\Vert ^{2} \bigr] \\ & \qquad{}\times\biggl[ \int _{0}^{1} \int _{\Omega } \int _{\tau _{1}}^{\tau _{2}} \varrho \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\vert z_{t}^{k} ( x,\varrho,\rho,t ) \bigr\vert ^{2}\,d\varrho \,d\rho\, dx\\ &\qquad{}+ \frac{1}{2} \int _{0}^{1} \int _{\tau _{1}}^{\tau _{2}} \int _{\Omega } \varrho \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\vert y_{t}^{k} ( x,\varrho,\rho,t ) \bigr\vert ^{2}\,d \varrho \,d\rho\, dx \biggr] \\ &\qquad{}+ \biggl( k_{1}- \biggl( \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \,d\varrho \biggr) \biggr) \int _{0}^{t} \bigl\Vert u_{tt}^{k} (s ) \bigr\Vert ^{2}\,ds\\ &\qquad{}+ \biggl( k_{2}- \biggl( \int _{\tau _{1}}^{\tau _{2}} \bigl\vert \mu _{2} ( \varrho ) \bigr\vert \,d\varrho \biggr) \biggr) \int _{0}^{t} \bigl\Vert v_{tt}^{k} (s ) \bigr\Vert ^{2}\,ds \\ &\quad\leq \bigl[ \bigl\Vert u_{tt}^{k} (0 ) \bigr\Vert ^{2}+ \mu \bigl\Vert \nabla u_{t}^{k} (0 ) \bigr\Vert ^{2}+ ( \lambda +\mu ) \bigl\Vert \operatorname{div} u_{t}^{k} (0 ) \bigr\Vert ^{2}+ \bigl\Vert v_{tt}^{k} (0 ) \bigr\Vert ^{2}\\ &\qquad{}+\mu \bigl\Vert \nabla v_{t}^{k} (0 ) \bigr\Vert ^{2}+ ( \lambda +\mu ) \bigl\Vert \operatorname{div} v_{t}^{k} (0 ) \bigr\Vert ^{2} \bigr] \\ &\qquad{}\times \biggl[ \int _{0}^{1} \int _{\Omega } \int _{\tau _{1}}^{\tau _{2}} \varrho \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\vert z_{t}^{k} ( x,\varrho,\rho,0 ) \bigr\vert ^{2}\,d\varrho \,d\rho\, dx\\ &\qquad{}+ \frac{1}{2} \int _{0}^{1} \int _{\tau _{1}}^{\tau _{2}} \int _{\Omega } \varrho \bigl\vert \mu _{1} ( \varrho ) \bigr\vert \bigl\vert y_{t}^{k} ( x,\varrho,\rho,0 ) \bigr\vert ^{2}\,d \varrho \,d\rho\, dx \biggr] \\ &\qquad{}+ \int _{0}^{t} \bigl\Vert u_{tt}^{k} ( s ) \bigr\Vert ^{2}\,ds + \int _{0}^{t} \bigl\Vert v_{tt}^{k} (s ) \bigr\Vert ^{2}\,ds+c. \end{aligned}$$
Taking
$$\begin{aligned} y^{k} ( t ) = \bigl\Vert u_{tt}^{k} ( t ) \bigr\Vert ^{2}+ \bigl\Vert v_{tt}^{k} ( t ) \bigr\Vert ^{2} \end{aligned}$$
and by using Gronwall’s inequality, we conclude that
$$\begin{aligned} &u_{t}^{k}\text{ and } v_{t}^{k} \text{ are bounded in }L^{ \infty } \bigl( 0,T;H_{0}^{1} ( \Omega ) \bigr), \end{aligned}$$
(3.40)
$$\begin{aligned} & u_{tt}^{k}\text{ and }v_{tt}^{k} \text{ are bounded in } L^{ \infty } \bigl( 0,T;L^{2} ( \Omega ) \bigr), \end{aligned}$$
(3.41)
$$\begin{aligned} &z_{t}^{k}\text{ and }y_{t}^{k} \text{ are bounded in } L^{ \infty } \bigl( 0,T;L^{2} \bigl( \Omega \times ( 0,1 ) \times ( \tau _{1},\tau _{2} ) \bigr) \bigr). \end{aligned}$$
(3.42)
Applying Dunford–Pettis’ theorem, we deduce from (3.13), (3.14), (3.15), (3.40), (3.41), and (3.42), replacing the sequence \(u_{k}\) with a subsequence if necessary, that
$$\begin{aligned} &u^{k}\rightarrow u, v^{k}\rightarrow v \text{ weak-star in } L^{ \infty } \bigl( 0,T;H_{0}^{1} ( \Omega ) \bigr), \end{aligned}$$
(3.43)
$$\begin{aligned} &z^{k}\rightarrow z, y^{k}\rightarrow y\text{ weak-star in } L^{ \infty } \bigl( 0,T; L^{2} \bigl(\Omega \times ( 0,1 ) \times ( \tau _{1},\tau _{2} ) \bigr) \bigr), \end{aligned}$$
(3.44)
$$\begin{aligned} &u_{t}^{k}\rightarrow u_{t}, v_{t}^{k}\rightarrow v_{t} \text{ weak-star in }L^{\infty } \bigl( 0,T;H_{0}^{1} ( \Omega ) \bigr), \end{aligned}$$
(3.45)
$$\begin{aligned} & u_{tt}^{k}\rightarrow u_{tt}, v_{tt}^{k}\rightarrow v_{tt} \text{ weak-star in } L^{\infty } \bigl( 0,T;L^{2} ( \Omega ) \bigr), \end{aligned}$$
(3.46)
$$\begin{aligned} &z_{t}^{k}\rightarrow z_{t}, y_{t}^{k}\rightarrow y_{t} \text{ weak-star in } L^{\infty } \bigl( 0,T;L^{2} \bigl( \Omega \times ( 0,1 )\times ( \tau _{1},\tau _{2} ) \bigr) \bigr). \end{aligned}$$
(3.47)
Corollary 3.3
The sequences of approximate solutions \(\lbrace u_{k},u_{k} \rbrace \) satisfy, as \(k\rightarrow \infty \),
$$\begin{aligned} \textstyle\begin{cases} f_{1} ( u_{k},v_{k} )\rightarrow f_{1} ( u,v )\textit{ strongly in } L^{\infty } ( 0,T;L^{2} ( \Omega ) ), \\ f_{2} ( u_{k},v_{k} )\rightarrow f_{2} ( u,v ) \textit{ strongly in } L^{\infty } ( 0,T;L^{2} ( \Omega ) ). \end{cases}\displaystyle \end{aligned}$$
(3.48)
Proof
The proof is similar to that of [11]. □
We can complete the proof of theorem as in [2].