Pullback attractor for N-dimensional thermoelastic coupled structure equations

Abstract

In this paper, proving the pullback asymptotic compactness of processes by the aid of a contractive function in space $X_{0}$, we prove the existence of a pullback attractor for N-dimensional nonautonomous thermoelastic coupled structure equations

$$\begin{aligned} &u_{tt}+\alpha\triangle^{2}u-\biggl[\beta+\sigma\biggl( \int_{\Omega}(\nabla u)^{2}\,dx\biggr)\biggr]\triangle u+ \gamma\triangle\theta+g(u)+\eta u_{t}=h(x,t), \\ &\quad \mbox{in } \Omega \times [\tau,\infty), \\ &\theta_{t}-\triangle\theta-\gamma\triangle u_{t}=q(x,t) \quad \mbox{in } \Omega\times [\tau,\infty), \end{aligned}$$

with the lateral load distribution function $h(x,t)$ and the external heat supply function $q(x,t)$ unnecessarily bounded. The nonlinear source term $g(u)$ is essentially $k_{1}(u+\frac{|u|^{\rho-1}u}{\rho+1})$ $(k_{1}>0)$ with $1<\rho\leq\frac{N}{N-2}$ if $N\geq 3$ and $1<\rho<\infty$ if $N=1,2$.

1 Introduction

In this paper, we consider the pullback asymptotic behavior of the following nonautonomous thermoelastic coupled structure equations:

$$\begin{aligned} &u_{tt}+\alpha\triangle^{2}u-\biggl[\beta+ \sigma\biggl( \int_{\Omega}(\nabla u)^{2}\,dx\biggr)\biggr]\triangle u+ \gamma\triangle\theta+g(u)+\eta u_{t}=h(x,t)\quad \mbox{in } \Omega \times [\tau,\infty), \end{aligned}$$

(1.1)

$$\begin{aligned} &\theta_{t}-\triangle\theta-\gamma\triangle u_{t}=q(x,t)\quad \mbox{in } \Omega\times [\tau,\infty), \end{aligned}$$

(1.2)

in a bounded domain $\Omega\subset R^{N}$ with smooth boundary. Here $\alpha,\beta,\gamma,\eta$ are all positive constants, which arise from a model of the nonlinear thermoelastic coupled vibration structure with clamped ends for simultaneously considering the medium damping, the viscous effect, and the nonlinear constitutive relation and thermoelasticity based on a theory of non-Fourier heat flux. The system is supplemented with the boundary conditions

$$\begin{aligned} & u(x,t)|_{\partial\Omega}=\frac{\partial u}{\partial v}(x,t)|_{\partial\Omega}=0, \qquad \theta(x,t)|_{\partial\Omega}=0,\quad t\geq\tau, \end{aligned}$$

(1.3)

for every $t>0$, and the initial conditions

$$\begin{aligned} u(x,\tau)=u_{0}(x),\qquad u_{t}(x, \tau)=v_{0}(x),\qquad \theta(x,\tau)=\theta_{0}(x),\quad x\in \Omega, \end{aligned}$$

(1.4)

where $u_{0}(x)$, $v_{0}(x)$ and $\theta_{0}(x)$ are assigned initial value functions.

Here the unknown variables $u(x,t)$ and $\theta(x,t)$ represent the vertical deflection of the structure and vertical component of the temperature gradient, respectively. The subscript t denotes the derivative with respect to t, $\sigma(\cdot)$ is the nonlinearity of the material and continuous nonnegative nonlinear real function, $g(u)$ is the source term, $h(x,t)$ is the lateral load distribution, and $q(x,t)$ is the external heat supply. Moreover, the source term $g(u)$ is essentially $k_{1}(u+\frac{|u|^{\rho-1}u}{\rho+1})$ ($k_{1}>0$) with $1<\rho\leq\frac{N}{N-2}$ if $N\geq 3$ and $1<\rho<\infty$ if $N=1,2$. Assumptions on nonlinear functions $\sigma(\cdot)$, $g(\cdot)$ and the external force function $h(x,t),q(x,t)$ will be specified later.

It is well known the global attractor on autonomous thermoelastic coupled structure equations has been considered in many papers. We refer the reader to [1–4] and the references therein.

However, in the actual life, the real systems are mostly nonautonomous. Recently, the nonautonomous infinite-dimensional dynamical system attracted attention of many people. For example, Chepyzhov and Vishik [5] firstly extended the notion of global attractor in the autonomous case to the nonautonomous case, which led to the concept of a uniform attractor. But the uniform attractor [6] was not applicable to nonautonomous systems with possibly unbounded trajectories as time increases to infinity (see [7–12]). To handle such problems, some new concepts and theories were brought up for nonautonomous case, and thus the pullback attractors were developed in [13–17], and they are a useful tool in understanding the dynamics of nonautonomous dynamical systems.

In this paper, we use the concept of pullback asymptotic compactness given in [7], and we prove the pullback asymptotic compactness by the method in [14] for nonautonomous system (1.1)-(1.4). Our fundamental assumptions on $\sigma(\cdot)$, $g(\cdot)$, $h(x,t)$, and $q(x,t)$ are given as follows.

Assumption 1

We assume that $\sigma(\cdot)\in C^{1}(R)$ satisfy

$$\begin{aligned} \sigma(z)z\geq\hat{\sigma}(z)\geq0,\quad \forall z\geq0, \end{aligned}$$

(1.5)

where $\hat{\sigma}(z)=\int_{0}^{z}\sigma(z)\,dz$. This condition is promptly satisfied if $\sigma(\cdot)$ is nondecreasing with $\sigma(0)=0$.

Assumption 2

The nonlinear term $g(\cdot)$ is a $C^{1}(R,R)$ function satisfying the following assumptions:

(H₁):

There exists a constant $k_{2}$ such that

$$\begin{aligned} \bigl\vert g(u)-g(v) \bigr\vert \leq k_{2} \vert u-v \vert \bigl(1+ \vert u \vert ^{\rho-1}+ \vert v \vert ^{\rho-1}\bigr). \end{aligned}$$

(1.6)

(H₂):

If $\hat{g}(s)$ is the primitive of $g(s)$, that is, $\hat{g}(s)=\int_{0}^{s} g(\tau)\,d\tau$, then

$$\begin{aligned} \liminf_{|s|\rightarrow\infty}\frac{\hat{g}(s)}{s^{2}}\geq0, \end{aligned}$$

(1.7)

and there exists a constant $k_{3}$ such that

$$\begin{aligned} \bigl\vert \hat{g}(u)-\hat{g}(v) \bigr\vert \leq k_{3}\bigl(u+v+ \vert u \vert ^{\rho}+ \vert v \vert ^{\rho}\bigr) \vert u-v \vert ,\quad \forall u,v\in R. \end{aligned}$$

(1.8)

(H₃):

There exists a constant $C_{0}\geq1$ such that

$$\begin{aligned} \liminf_{|s|\rightarrow\infty}\frac{sg(s)-C_{0}\hat{g}(s)}{s^{2}}\geq0. \end{aligned}$$

(1.9)

Assumption 3

Functions $h(x,t)$ and $q(x,t)$ for $t\in R$, $x\in\Omega$ are locally square integrable in time, that is, $h(x,t), q(x,t)\in L^{2}_{\mathrm{loc}}(R,L^{2}(\Omega))$, and for any $t\in R$,

$$\begin{aligned} \int_{-\infty}^{t}e^{\delta s} \bigl\Vert h(x,s) \bigr\Vert ^{2} \,ds< \infty \end{aligned}$$

(1.10)

and

$$\begin{aligned} \int_{-\infty}^{t}e^{\delta s} \bigl\Vert q(x,s) \bigr\Vert ^{2} \,ds< \infty, \end{aligned}$$

(1.11)

where $\delta>0$ is a small real number, which will be characterized later.

Under these assumptions, we prove the existence of a pullback attractor for nonautonomous thermoelastic coupled structure equation system (1.1)-(1.4).

2 Preliminaries

We first introduce the following abbreviations:

$$H=L^{2}(\Omega), \qquad \Vert \cdot \Vert = \Vert \cdot \Vert _{L^{2}(\Omega)}. $$

Let $(\cdot,\cdot)$ denote the H-inner product, and let $\Vert \nabla\cdot \Vert$ and $\Vert \triangle\cdot \Vert$ be the norms of $H_{0}^{1}(\Omega)$ and $H_{0}^{2}(\Omega)$, respectively.

We denote the space

$$X_{0}=H_{0}^{2}(\Omega)\times L^{2}( \Omega)\times L^{2}(\Omega) $$

equipped with the norm

$$\Vert \cdot \Vert ^{2}_{X_{0}}= \Vert \triangle u \Vert ^{2}+ \Vert v \Vert ^{2}+ \Vert \theta \Vert ^{2}. $$

The sign $H_{1}\hookrightarrow\hookrightarrow H_{2}$ denotes compact embedding of $H_{1}$ into $H_{2}$. For brevity, we use the same letter C to denote different positive constants.

3 Abstract results

In this section, we recall some definitions and results concerning the pullback attractor for nonautonomous dynamical systems. These definitions and results can be found in [11–15] and the references therein.

Let $(X_{0},d)$ be a complete metric space, and let $(Q,\rho)$ be a metric space which is called the parameter space. We define a nonautonomous dynamical system by a cocycle mapping $\Phi:R_{+}\times Q\times X_{0}\rightarrow X_{0}$, which is driven by an autonomous dynamical system θ acting on a parameter space Q. Specifically, $\theta=\{\theta_{t}\}_{t\in R}$ is a dynamical system on Q, that is, it is a group of homeomorphisms under composition on Q with the properties that:

• (1) $\theta_{0}(q)=q$ for all $q\in Q$;

• (2) $\theta_{t+\tau}(q)=\theta_{t}(\theta_{\tau}(q))$ for all $t,\tau\in R$;

• (3) The mapping $(t,q)\rightarrow\theta_{t}(q)$ is continuous.

Definition 3.1

([13–18])

A mapping Φ is said to be a cocycle on $X_{0}$ with respect to group θ if

• (1) $\Phi(0,q,x)=x$ for all $(q,x)\in Q\times X_{0}$;

• (2) $\Phi(t+s,q,x)=\Phi(s,\theta_{t}(q),\Phi(t,q,x))$ for all $s,t\in R_{+}$ and all $(q,x)\in Q\times X_{0}$;

• (3) the mapping $\Phi(t,q,\cdot):X_{0}\rightarrow X_{0}$ is continuous for all $(t,q)\in R^{+}\times Q$.

Definition 3.2

([13–18])

A family of nonempty compact sets $A=\{A_{q}\}_{q\in Q}$ is said to be a pullback (or cocycle) attractor if, for each $q\in Q$, it satisfies

• (1) $\Phi(t,q,A_{q})=A_{\theta_{t}(q)}$ for all $t\in R^{+}$ (Φ-invariance);

• (2) $\lim_{t\rightarrow\infty}\operatorname{dist}(\Phi(t,\theta_{-t}(q), B),A_{q})=0$ for any bounded subset $B\subset X$ (pullback attracting).

Definition 3.3

([13–18])

A family $D=\{D_{q}\}_{q\in Q}\in K$ is said to be pullback absorbing if for each $q\in Q$ and any bounded subset B of $X_{0}$, there exists $t_{0}(q,B)\geq0$ such that

$$\begin{aligned} \Phi\bigl(t,\theta_{-t}(q), B_{\theta_{-t}(q)}\bigr) \subset D_{q} \quad \mbox{for all } t\geq t_{0}. \end{aligned}$$

(3.1)

Definition 3.4

([13])

Let $(\theta,\Phi)$ be a nonautonomous dynamical system on $Q\times X_{0}$, and let $D=\{D_{q}\}_{q\in Q}$ be a family of bounded subsets of $X_{0}$. The cocycle Φ is said to be pullback D-asymptotically compact if for any sequences $t_{n}\rightarrow\infty$ and $x_{n}\in D_{\theta_{-t_{n}}(q)}$, the sequence $\Phi(t_{n},\theta_{-t_{n}}(q),x_{n})$ is precompact in $X_{0}$.

Lemma 3.1

([17])

Let $(\theta,\Phi)$ be a nonautonomous dynamical system on $Q\times X_{0}$. Assume that the family $D=\{D_{q}\}_{q\in Q}$ is pullback absorbing for Φ and Φ is pullback D-asymptotically compact. Then Φ possesses attractor $A=\{A_{q}\}_{q\in Q}$, and

$$A_{q}=\bigcap_{t\geq0}\overline{\bigcup_{s\geq t}\Phi\bigl(s, \theta{-s}(q), D_{\theta_{-s}(q)}\bigr)},\quad q\in Q. $$

For this matter, first we give the following concept and lemma.

Definition 3.5

([17])

Let $X_{0}$ be a Banach space, and let B be a bounded subset of $X_{0}$. We call a function $\phi(\cdot,\cdot)$ defined on $X_{0}\times X_{0}$ a contractive function on $B\times B$ if for any sequence $\{x_{n}\}_{n\in N}\subset B$, there exists a subsequence $\{x_{nk}\}_{k\in N}\subset \{x_{n}\}_{n\in N}$ such that

$$\begin{aligned} \lim_{k\rightarrow\infty}\lim_{l\rightarrow\infty} \phi(x_{nk},x_{nl})=0. \end{aligned}$$

(3.2)

Lemma 3.2

([17])

Let $(\theta,\Phi)$ be a nonautonomous dynamical system on $Q\times X_{0}$. Suppose that bounded families $D=\{D_{q}\}_{q\in Q}$ and $\tilde{D}=\{\tilde{D}_{q}\}_{q\in Q}$ are such that, for any $q\in Q$, there exists $t_{q}=t(q,D,\tilde{D})\geq0$ such that

$$\begin{aligned} \Phi\bigl(t,\theta_{-t}(q), D_{\theta_{-t}(q)}\bigr) \subset\tilde{D}_{q}\quad \textit{for all } t\geq t_{q}. \end{aligned}$$

(3.3)

Assume that, for any $\varepsilon>0$ and $q\in Q$, there exist $t=t(\varepsilon,\tilde{D},q)\geq0$ and a contractive function $\phi_{t,q}(\cdot,\cdot)$ defined on $\tilde{D}_{\theta_{-t}(q)}\times\tilde{D}_{\theta_{-t}(q)}$ such that

$$\bigl\Vert \Phi\bigl(t,\theta_{-t}(q),x\bigr)-\Phi\bigl(t, \theta_{-t}(q),y\bigr) \bigr\Vert _{X_{0}}\leq \varepsilon+ \phi_{t,q}(x,y)\quad \textit{for all } x,y\in \tilde{D}_{\theta_{-t}(q)}, $$

where $\phi_{t,q}$ depends on $t,q$. Then Φ is pullback D-asymptotically compact in $X_{0}$.

4 Global solutions and pullback attracting set

Using the classical Galerkin method, we can establish our main theorem of this section on the existence and uniqueness of a global solution to problem (1.1)-(1.4).

Theorem 4.1

Assume that $h(x,t), q(x,t)\in L^{2}_{\mathrm{loc}}(R,L^{2}(\Omega))$ and that assumptions ($H_{1}$)-($H_{3}$) on the function $g(\cdot)$ hold. Then for any $(u_{0}, v_{0},\theta_{0})\in X_{0}$, problem (1.1)-(1.4) has a unique solution $(u,u_{t},\theta)$ satisfying $(u,u_{t},\theta)\in C^{0}(R_{\tau};X_{0})$, where $R_{\tau}=[\tau,\infty)$.

For simplicity, we write $y(r)=(u(r),\partial_{r}u(r), \theta(r))=(u(r),v(r),\theta(r))$, $y_{0}=(u_{0},v_{0},\theta_{0})$. We denote by $X_{0}$ the space of vector functions $y(r)=(u(r),v(r),\theta(r))$ with the norm $\Vert y \Vert ^{2}_{X_{0}}= \Vert \triangle u \Vert ^{2}+ \Vert v \Vert ^{2}+ \Vert \theta \Vert ^{2}$.

We can construct the nonautonomous dynamical system generated by problem (1.1)-(1.4) in $X_{0}$. We consider $Q=R$ and $\theta_{t}\tau=\tau+t$. Then we define

$$\begin{aligned} \Phi(t,\tau,y_{0})=y(t+\tau;\tau,y_{0})= \bigl(u(t+\tau),v(t+\tau),\theta(t+\tau)\bigr),\quad \tau\in R,t \geq0,y_{0}\in X_{0}. \end{aligned}$$

(4.1)

The uniqueness of a solution to problem (1.1)-(1.4) implies that

$$\Phi(t+s,\tau,y_{0})=\Phi\bigl(t,s+\tau,\Phi(s,\tau,y_{0}) \bigr),\quad \tau\in R, t,s\geq0, y_{0}\in X_{0}, $$

and, for all $\tau\in R, t\geq0$, the mapping $\Phi(t,\tau,\cdot):X_{0}\rightarrow X_{0}$ defined by (4.1) is continuous. Consequently, for any $(t,\tau)\in R^{+}\times R$, the mapping $\Phi(t,\tau,\cdot)$ defined by (4.1) is a continuous cocycle on $X_{0}$.

Another main result of this section is as follows.

Theorem 4.2

Suppose $\alpha>3\gamma$, $h(x,t)$ and $q(x,t)\in L^{2}_{\mathrm{loc}}(R;H)$ satisfy (1.10) and (1.11) with δ satisfying $0<\delta<\varepsilon_{0}$ $(0<\varepsilon_{0}\leq\min\{\frac{\sqrt{1+4\mu^{2}}-1}{2}, \frac{4\alpha\lambda^{2}}{6\eta+5}, \frac{\alpha}{3}, \sqrt{9+3\eta}-3, \frac{\alpha\lambda^{2}}{\eta}\})$. Then there exist a family of bounded sets $D=\{D_{q}\}_{q\in Q}$ in $X_{0}$ which is pullback absorbing for Φ defined by (4.1) and a family of bounded sets $\tilde{D}=\{\tilde{D}_{q}\}_{q\in Q}$ satisfying (3.3).

Proof

Let $t_{0}\in R$, $\tau\geq 0$, and $y_{0}=(u_{0},v_{0},\theta_{0})\in X_{0}$ be fixed. Define

$$\begin{aligned} &u(r)=u(r,t_{0}-\tau,u_{0}),\qquad v(r)=u'(r,t_{0}- \tau,v_{0}),\quad \mbox{and} \\ & \theta(r)=\theta(r,t_{0}-\tau,\theta_{0})\quad \mbox{for } r\geq t_{0}-\tau \end{aligned}$$

and

$$\bigl(u(r),v(r),\theta(r)\bigr)=\Phi(r-t_{0}+\tau,t_{0}- \tau,y_{0})\quad \mbox{for } r\geq t_{0}-\tau. $$

Multiplying equations (1.1) and (1.2) by $p=u_{t}+\varepsilon_{0} u$ and θ, respectively, and then summing, we obtain

$$\begin{aligned} &\frac{1}{2}\,\frac{d}{dr}\bigl[ \Vert p \Vert ^{2} +\alpha \Vert \triangle u \Vert ^{2}-\eta \varepsilon_{0} \Vert u \Vert ^{2}+\beta \Vert \nabla u \Vert ^{2} +\hat{\sigma}\bigl( \Vert \nabla u \Vert ^{2} \bigr)+ \Vert \theta \Vert ^{2}\bigr] \\ &\qquad {}-\varepsilon_{0} \Vert p \Vert ^{2}+ \varepsilon_{0}\alpha \Vert \triangle u \Vert ^{2}+\eta \Vert p \Vert ^{2}-\eta\varepsilon_{0}^{2} \Vert u \Vert ^{2} +\varepsilon_{0}\beta \Vert \nabla u \Vert ^{2}+\varepsilon_{0}\sigma\bigl( \Vert \nabla u \Vert ^{2}\bigr) \Vert \nabla u \Vert ^{2} \\ &\qquad {}+\varepsilon_{0}\gamma(\triangle u,\theta)+ \Vert \nabla \theta \Vert ^{2}+\varepsilon_{0}^{2}(u,p) + \bigl(g(u),p\bigr) \\ &\quad =(h,p)+(q,\theta). \end{aligned}$$

(4.2)

For simplicity, define $\phi(u)=\int_{\Omega}\hat{g}(u)\,dx$. By assumption (1.7) on $g(\cdot)$ it is obvious that $\phi(u)\geq0$. By assumption (1.9) on $g(\cdot)$ we have

$$\bigl(g(u),u\bigr)-C_{0}\phi(u)+\frac{\varepsilon_{0}}{4} \Vert u \Vert ^{2}\geq-M, $$

where $C_{0}\geq1$. So

$$\begin{aligned} \bigl(g(u),p\bigr)&=\bigl(g(u),u_{t}\bigr)+ \varepsilon_{0}\bigl(g(u),u\bigr) \\ &\geq \int_{\Omega}\,\frac{d}{dr}\hat{g}(u)\,dx+ \varepsilon_{0} C_{0}\phi(u)-\frac{\varepsilon_{0}^{2}}{4} \Vert u \Vert ^{2}-\varepsilon_{0} M \\ &\geq\frac{d}{dr}\phi(u)+\varepsilon_{0} C_{0}\phi(u)- \frac{\varepsilon_{0}^{2}}{4} \Vert u \Vert ^{2}-\varepsilon_{0} M. \end{aligned}$$

(4.3)

By Young’s inequality we have

$$\begin{aligned} \bigl\vert (h,p) \bigr\vert \leq\frac{1}{\eta} \Vert h \Vert ^{2}+\frac{\eta}{4} \Vert p \Vert ^{2} \end{aligned}$$

(4.4)

and

$$\begin{aligned} \bigl\vert (q,\theta) \bigr\vert \leq\frac{ \lambda_{0}^{2}}{2} \Vert \theta \Vert ^{2}+\frac{1}{2\lambda_{0}^{2}} \Vert q \Vert ^{2}\leq\frac{1}{2} \Vert \nabla \theta \Vert ^{2}+\frac{1}{2\lambda_{0}^{2}} \Vert q \Vert ^{2}, \end{aligned}$$

(4.5)

where $\lambda_{0}$ is the first eigenvalue of ∇ in $L^{2}(\Omega)$.

By (4.3)-(4.5) from (4.2) we have

$$\begin{aligned} &\frac{1}{2}\,\frac{d}{dr}\bigl[ \Vert p \Vert ^{2} +\alpha \Vert \triangle u \Vert ^{2}-\eta \varepsilon_{0} \Vert u \Vert ^{2}+\beta \Vert \nabla u \Vert ^{2} +\hat{\sigma}\bigl( \Vert \nabla u \Vert ^{2} \bigr)+ \Vert \theta \Vert ^{2}+2\phi(u)\bigr] \\ &\qquad {}-\varepsilon_{0} \Vert p \Vert ^{2}+ \varepsilon_{0}\alpha \Vert \triangle u \Vert ^{2}+ \frac{3\eta}{4} \Vert p \Vert ^{2}-\eta\varepsilon_{0}^{2} \Vert u \Vert ^{2} +\varepsilon_{0}\beta \Vert \nabla u \Vert ^{2}+\varepsilon_{0}\sigma\bigl( \Vert \nabla u \Vert ^{2}\bigr) \Vert \nabla u \Vert ^{2} \\ &\qquad {} +\varepsilon_{0}\gamma(\triangle u,\theta)+ \frac{1}{2} \Vert \nabla\theta \Vert ^{2}+ \varepsilon_{0}^{2}(u,p) +\varepsilon_{0} C_{0}\phi(u)- \frac{\varepsilon_{0}^{2}}{4} \Vert u \Vert ^{2} \\ &\quad \leq\frac{1}{\eta} \Vert h \Vert ^{2}+\frac{1}{2\lambda_{0}^{2}} \Vert q \Vert ^{2}+\varepsilon_{0} M. \end{aligned}$$

(4.6)

By Young’s inequality we have

$$\begin{aligned} \varepsilon_{0}\gamma(\triangle u,\theta)\geq- \frac{\varepsilon_{0}\gamma}{2} \Vert \triangle u \Vert ^{2}-\frac{\varepsilon_{0}\gamma}{2} \Vert \theta \Vert ^{2} \end{aligned}$$

(4.7)

and

$$\begin{aligned} \varepsilon_{0}^{2}(u,p)\geq- \varepsilon_{0}^{2}\biggl( \Vert u \Vert ^{2}+ \frac{1}{4} \Vert p \Vert ^{2}\biggr). \end{aligned}$$

(4.8)

So

$$\begin{aligned} &\biggl(\frac{3\eta}{4}-\varepsilon_{0}\biggr) \Vert p \Vert ^{2}+\varepsilon_{0}\alpha \Vert \triangle u \Vert ^{2}-\eta\varepsilon_{0}^{2} \Vert u \Vert ^{2} +\varepsilon_{0}\beta \Vert \nabla u \Vert ^{2}+\varepsilon_{0}\sigma\bigl( \Vert \nabla u \Vert ^{2}\bigr) \Vert \nabla u \Vert ^{2} \\ &\qquad {} +\varepsilon_{0}\gamma(\triangle u,\theta)+\frac{1}{2} \Vert \nabla\theta \Vert ^{2}+\varepsilon_{0}^{2}(u,p) +\varepsilon_{0} C_{0}\phi(u)- \frac{\varepsilon_{0}^{2}}{4} \Vert u \Vert ^{2} \\ &\quad \geq \biggl(\frac{3\eta}{4}-\varepsilon_{0}-\frac{\varepsilon_{0}^{2}}{4} \biggr) \Vert p \Vert ^{2}+\biggl(\varepsilon_{0}\alpha- \frac{\varepsilon_{0}\gamma}{2}\biggr) \Vert \triangle u \Vert ^{2} +\biggl(-\eta \varepsilon_{0}^{2}-\frac{5\varepsilon_{0}^{2}}{4}\biggr) \Vert u \Vert ^{2} \\ &\qquad {}+\frac{1}{2} \Vert \nabla\theta \Vert ^{2}- \frac{\varepsilon_{0}\gamma}{2} \Vert \theta \Vert ^{2}+\varepsilon_{0} \beta \Vert \nabla u \Vert ^{2}+\varepsilon_{0}\sigma \bigl( \Vert \nabla u \Vert ^{2}\bigr) \Vert \nabla u \Vert ^{2} +\varepsilon_{0} C_{0}\phi(u) \\ &\quad \geq \biggl(\frac{3\eta}{4}-\varepsilon_{0}-\frac{\varepsilon_{0}^{2}}{4} \biggr) \Vert p \Vert ^{2} +\biggl(\frac{2\varepsilon_{0}\alpha}{3}- \frac{\varepsilon_{0}\gamma}{2}\biggr) \Vert \triangle u \Vert ^{2} +\biggl( \frac{\varepsilon_{0}\alpha\lambda^{2}}{3}-\eta\varepsilon_{0}^{2}-\frac{5\varepsilon_{0}^{2}}{4} \biggr) \Vert u \Vert ^{2} \\ &\qquad {}+\biggl(\frac{\lambda_{0}^{2}}{2}-\frac{\varepsilon_{0}\gamma}{2}\biggr) \Vert \theta \Vert ^{2}+\varepsilon_{0}\beta \Vert \nabla u \Vert ^{2}+\varepsilon_{0}\sigma\bigl( \Vert \nabla u \Vert ^{2}\bigr) \Vert \nabla u \Vert ^{2} + \varepsilon_{0} C_{0}\phi(u), \end{aligned}$$

where λ is the first eigenvalue of △ in $L^{2}(\Omega)$. Let

$$\begin{aligned} L(u,p,\theta)=\frac{1}{2}\bigl( \Vert p \Vert ^{2} +\alpha \Vert \triangle u \Vert ^{2}-\eta\varepsilon_{0} \Vert u \Vert ^{2}+\beta \Vert \nabla u \Vert ^{2} +\hat{ \sigma}\bigl( \Vert \nabla u \Vert ^{2}\bigr)+ \Vert \theta \Vert ^{2}+2\phi(u)\bigr)\geq0 \end{aligned}$$

and

$$\begin{aligned} Y(u,p,\theta)={}&\biggl(\frac{3\eta}{4}-\varepsilon_{0}- \frac{\varepsilon_{0}^{2}}{4}\biggr) \Vert p \Vert ^{2} +\biggl( \frac{2\varepsilon_{0}\alpha}{3}-\frac{\varepsilon_{0}\gamma}{2}\biggr) \Vert \triangle u \Vert ^{2}\\ &{} +\biggl(\frac{\varepsilon_{0}\alpha\lambda^{2}}{3}-\eta\varepsilon_{0}^{2}- \frac{5\varepsilon_{0}^{2}}{4}\biggr) \Vert u \Vert ^{2} +\biggl(\frac{\lambda_{0}^{2}}{2}-\frac{\varepsilon_{0}\gamma}{2}\biggr) \Vert \theta \Vert ^{2}\\ &{}+\varepsilon_{0}\beta \Vert \nabla u \Vert ^{2}+\varepsilon_{0}\sigma\bigl( \Vert \nabla u \Vert ^{2}\bigr) \Vert \nabla u \Vert ^{2} + \varepsilon_{0} C_{0}\phi(u). \end{aligned}$$

Then considering $0<\varepsilon_{0}\leq\min\{\frac{\lambda_{0}^{2}}{\gamma+1}, \frac{4\alpha\lambda^{2}}{6\eta+15}, \sqrt{9+3\eta}-3 , \frac{\alpha\lambda^{2}}{\eta} \}$, $\alpha>3\gamma$, and $C_{0}\geq1$, from (1.5) we get

$$\begin{aligned} &Y(u,p,\theta)-\varepsilon_{0} L(u,p,\theta) \\ &\quad \geq\biggl(\frac{3\eta}{4}-\frac{3\varepsilon_{0}}{2}-\frac{\varepsilon_{0}^{2}}{4}\biggr) \Vert p \Vert ^{2} +\biggl(\frac{2\varepsilon_{0}\alpha}{3}-\frac{\varepsilon_{0}\gamma}{2}- \frac{\varepsilon_{0}\alpha}{2}\biggr) \Vert \triangle u \Vert ^{2} \\ &\qquad {}+\biggl( \frac{\varepsilon_{0}\alpha\lambda^{2}}{3}-\frac{\eta\varepsilon_{0}^{2}}{2}-\frac{5\varepsilon_{0}^{2}}{4}\biggr) \Vert u \Vert ^{2} +\biggl(\frac{\lambda_{0}^{2}}{2}-\frac{\varepsilon_{0}\gamma}{2}-\frac{\varepsilon_{0}}{2}\biggr) \Vert \theta \Vert ^{2} \\ &\qquad {}+\varepsilon_{0}\sigma\bigl( \Vert \nabla u \Vert ^{2}\bigr) \Vert \nabla u \Vert ^{2}- \varepsilon_{0}\hat{\sigma}\bigl( \Vert \nabla u \Vert ^{2}\bigr) +\varepsilon_{0} (C_{0}-1)\phi(u) \geq 0. \end{aligned}$$

From (4.6) we have

$$\begin{aligned} \frac{d}{dr}L(u,p,\theta)+\varepsilon_{0} L(u,p, \theta)\leq \frac{1}{\eta} \Vert h \Vert ^{2}+\frac{1}{2\lambda_{0}^{2}} \Vert q \Vert ^{2}+\varepsilon_{0} M. \end{aligned}$$

(4.9)

Note that

$$\frac{d}{dr}e^{\delta r}L(u,p,\theta) =\delta e^{\delta r}L(u,p, \theta) +e^{\delta r}\,\frac{d}{dr}L(u,p,\theta), $$

so by (4.9) we have

$$\begin{aligned} \frac{d}{dr}\bigl[e^{\delta r}L(u,p,\theta)\bigr] \leq{}&( \delta-\varepsilon_{0}) e^{\delta r}L(u,p,\theta) \\ &{}+e^{\delta r} \biggl(\frac{1}{\eta} \Vert h \Vert ^{2}+\frac{1}{2\lambda_{0}^{2}} \Vert q \Vert ^{2}+\varepsilon_{0} M\biggr). \end{aligned}$$

(4.10)

By integrating (4.10) over the interval $[t_{0}-\tau,t_{0}]$, with $L(u,p,\theta)\geq0$, we obtain

$$\begin{aligned} e^{\delta t_{0}}L\bigl(u(t_{0}),p(t_{0}), \theta(t_{0})\bigr) \leq &e^{\delta (t_{0}-\tau)}L\bigl(u(t_{0}- \tau),p(t_{0}-\tau),\theta(t_{0}-\tau)\bigr) \\ &{}+(\delta-\varepsilon_{0}) \int_{t_{0}-\tau}^{t_{0}}e^{\delta s}L\bigl(u(s),p(s), \theta(s)\bigr)\,ds \\ &{}+ \int_{t_{0}-\tau}^{t_{0}} e^{\delta s}\biggl( \frac{1}{\eta} \Vert h \Vert ^{2}+\frac{1}{2\lambda_{0}^{2}} \Vert q \Vert ^{2}\biggr)\,ds \\ &{}+\frac{\varepsilon_{0} M}{\delta}\bigl(e^{\delta t_{0}}-e^{\delta(t_{0}-\tau)} \bigr) . \end{aligned}$$

(4.11)

Since $\delta<\varepsilon_{0}$, from (4.11) we have

$$\begin{aligned} & \bigl\Vert p(t_{0}) \bigr\Vert ^{2} + \alpha \bigl\Vert \triangle u(t_{0}) \bigr\Vert ^{2}-\eta \varepsilon_{0} \bigl\Vert u(t_{0}) \bigr\Vert ^{2}+\beta \bigl\Vert \nabla u(t_{0}) \bigr\Vert ^{2} \\ &\qquad {}+\hat{\sigma}\bigl( \bigl\Vert \nabla u(t_{0}) \bigr\Vert ^{2}\bigr)+ \bigl\Vert \theta(t_{0}) \bigr\Vert ^{2}+2\phi\bigl(u(t_{0})\bigr) \\ &\quad \leq e^{-\delta \tau}\bigl( \bigl\Vert p(t_{0}-\tau) \bigr\Vert ^{2} +\alpha \bigl\Vert \triangle u(t_{0}-\tau) \bigr\Vert ^{2}-\eta\varepsilon_{0} \bigl\Vert u(t_{0}-\tau) \bigr\Vert ^{2}+\beta \bigl\Vert \nabla u(t_{0}-\tau) \bigr\Vert ^{2} \\ &\qquad {}+\hat{\sigma}\bigl( \bigl\Vert \nabla u(t_{0}-\tau) \bigr\Vert ^{2}\bigr)+ \bigl\Vert \theta(t_{0}-\tau) \bigr\Vert ^{2}+2\phi\bigl(u(t_{0}-\tau)\bigr)\bigr) \\ &\qquad {}+e^{-\delta t_{0}} \int_{t_{0}-\tau}^{t_{0}} e^{\delta s}\biggl( \frac{2}{\eta} \Vert h \Vert ^{2}+\frac{1}{\lambda_{0}^{2}} \Vert q \Vert ^{2}\biggr)\,ds +\frac{2\varepsilon_{0} M}{\delta}\bigl(1-e^{-\delta\tau} \bigr) . \end{aligned}$$

(4.12)

If we take $C_{1}=\max\{2,1+\frac{2\varepsilon_{0}^{2}}{\lambda^{2}}\}$, then since $\Vert u \Vert ^{2}\leq \frac{1}{\lambda^{2}} \Vert \triangle u \Vert ^{2}$, we have

$$\begin{aligned} & \bigl\Vert \triangle u(t_{0}) \bigr\Vert ^{2}+ \bigl\Vert v(t_{0}) \bigr\Vert ^{2}+ \bigl\Vert \theta(t_{0}) \bigr\Vert ^{2} \\ &\quad \leq \bigl\Vert \triangle u(t_{0}) \bigr\Vert ^{2}+ \bigl\Vert v(t_{0})+\varepsilon_{0} u(t_{0})-\varepsilon_{0} u(t_{0}) \bigr\Vert ^{2}+ \bigl\Vert \theta(t_{0}) \bigr\Vert ^{2} \\ &\quad \leq \bigl\Vert \triangle u(t_{0}) \bigr\Vert ^{2}+2 \bigl\Vert p(t_{0}) \bigr\Vert ^{2}+2\varepsilon_{0}^{2} \bigl\Vert u(t_{0}) \bigr\Vert ^{2}+ \bigl\Vert \theta(t_{0}) \bigr\Vert ^{2} \\ &\quad \leq \biggl(1+\frac{2\varepsilon_{0}^{2}}{\lambda^{2}}\biggr) \bigl\Vert \triangle u(t_{0}) \bigr\Vert ^{2}+2 \bigl\Vert p(t_{0}) \bigr\Vert ^{2}+ \bigl\Vert \theta(t_{0}) \bigr\Vert ^{2} \\ &\quad = C_{1}\bigl( \bigl\Vert \triangle u(t_{0}) \bigr\Vert ^{2}+ \bigl\Vert p(t_{0}) \bigr\Vert ^{2}+ \bigl\Vert \theta(t_{0}) \bigr\Vert ^{2}\bigr). \end{aligned}$$

(4.13)

On the other hand, setting $C_{2}=\min\{1,\alpha-\frac{\eta\varepsilon_{0}}{\lambda^{2}}\}$, we obtain

$$\begin{aligned} \begin{aligned}[b] &L\bigl(u(t_{0}),p(t_{0}),\theta(t_{0}) \bigr)\\ &\quad \geq \biggl( \bigl\Vert p(t_{0}) \bigr\Vert ^{2} + \biggl(\alpha-\frac{\eta\varepsilon_{0}}{\lambda^{2}}\biggr) \bigl\Vert \triangle u(t_{0}) \bigr\Vert ^{2}+\beta \bigl\Vert \nabla u(t_{0}) \bigr\Vert ^{2} \\ &\qquad {} +\hat{\sigma}\bigl( \bigl\Vert \nabla u(t_{0}) \bigr\Vert ^{2}\bigr)+ \bigl\Vert \theta(t_{0}) \bigr\Vert ^{2}+2\phi\bigl(u(t_{0})\bigr)\biggr) \\ &\quad \geq C_{2}\bigl( \bigl\Vert p(t_{0}) \bigr\Vert ^{2}+ \bigl\Vert \triangle u(t_{0}) \bigr\Vert ^{2}+ \bigl\Vert \theta(t_{0}) \bigr\Vert ^{2} \bigr). \end{aligned} \end{aligned}$$

(4.14)

So from (4.13)-(4.14) we get

$$\begin{aligned} & \bigl\Vert \triangle u(t_{0}) \bigr\Vert ^{2}+ \bigl\Vert v(t_{0}) \bigr\Vert ^{2}+ \bigl\Vert \theta(t_{0}) \bigr\Vert ^{2} \\ &\quad \leq C_{1}\bigl( \bigl\Vert \triangle u(t_{0}) \bigr\Vert ^{2}+ \bigl\Vert p(t_{0}) \bigr\Vert ^{2}+ \bigl\Vert \theta(t_{0}) \bigr\Vert ^{2}\bigr) \\ &\quad \leq \frac{C_{1}}{C_{2}}L\bigl(u(t_{0}),p(t_{0}), \theta(t_{0})\bigr) \\ &\quad \leq \frac{C_{1}}{C_{2}}\biggl\{ e^{-\delta \tau}\bigl( \bigl\Vert p(t_{0}-\tau) \bigr\Vert ^{2}+\alpha \bigl\Vert \triangle u(t_{0}-\tau) \bigr\Vert ^{2}-\eta\varepsilon_{0} \bigl\Vert u(t_{0}-\tau) \bigr\Vert ^{2}+\beta \bigl\Vert \nabla u(t_{0}-\tau) \bigr\Vert ^{2} \\ &\qquad {}+\hat{\sigma}\bigl( \bigl\Vert \nabla u(t_{0}-\tau) \bigr\Vert ^{2}\bigr)+ \bigl\Vert \theta(t_{0}-\tau) \bigr\Vert ^{2}+2\phi\bigl(u(t_{0}-\tau)\bigr)\bigr) \\ &\qquad {}+e^{-\delta t_{0}} \int_{t_{0}-\tau}^{t_{0}} e^{\delta s}\biggl( \frac{2}{\eta} \Vert h \Vert ^{2}+\frac{1}{\lambda_{0}^{2}} \Vert q \Vert ^{2}\biggr)\,ds +\frac{2\varepsilon_{0} M}{\delta}\bigl(1-e^{-\delta\tau} \bigr)\biggr\} . \end{aligned}$$

(4.15)

Let $\hat{D}_{\delta,X_{0}}$ ($\hat{D}_{\delta,X_{0}}$ denotes the class of all families $D=\{D_{t}\}_{t\in R}$) be given. For all $y(t_{0}-\tau)=y_{0}\in D(t_{0}-\tau)$, $t\in R$, and $\tau\geq0$, from assumption (1.8) on $\hat{g}(\cdot)$ we know that $\phi(u(t_{0}-\tau))$ is bounded. Using the midvalue theorem of integration, from the assumption that $\sigma(\cdot)\in C^{1}(R)$ we have that $\hat{\sigma}( \Vert \nabla u(t_{0}-\tau) \Vert ^{2})$ is bounded, too. So from (4.15) we easily obtain

$$\begin{aligned} & \bigl\Vert \Phi(\tau,t_{0}-\tau,y_{0}) \bigr\Vert ^{2}_{X_{0}} \\ &\quad \leq \frac{C_{1}}{C_{2}}\biggl\{ e^{-\delta \tau}\bigl( \bigl\Vert p(t_{0}-\tau) \bigr\Vert ^{2} +\alpha \bigl\Vert \triangle u(t_{0}-\tau) \bigr\Vert ^{2}-\eta \varepsilon_{0} \bigl\Vert u(t_{0}-\tau) \bigr\Vert ^{2}+\beta \bigl\Vert \nabla u(t_{0}-\tau) \bigr\Vert ^{2} \\ &\qquad {}+\hat{\sigma}\bigl( \bigl\Vert \nabla u(t_{0}-\tau) \bigr\Vert ^{2}\bigr)+ \bigl\Vert \theta(t_{0}-\tau) \bigr\Vert ^{2}+2\phi\bigl(u(t_{0}-\tau)\bigr)\bigr) \\ &\qquad {}+e^{-\delta t_{0}} \int_{-\infty}^{t_{0}} e^{\delta s}\biggl( \frac{2}{\eta} \Vert h \Vert ^{2}+\frac{1}{\lambda_{0}^{2}} \Vert q \Vert ^{2}\biggr)\,ds +\frac{2\varepsilon_{0} M}{\delta}\bigl(1-e^{-\delta\tau} \bigr)\biggr\} \end{aligned}$$

(4.16)

for all $y_{0}\in D(t_{0}-\tau)$, $t_{0}\in R$, and $\tau\geq0$. Set

$$\begin{aligned} (R_{t})^{2}=2\frac{C_{1}}{C_{2}} e^{-\delta t_{0}} \int_{-\infty}^{t}e^{\delta s}\biggl( \frac{2}{\eta} \bigl\Vert h(s) \bigr\Vert ^{2}+ \frac{1}{\lambda_{0}^{2}} \bigl\Vert q(s) \bigr\Vert ^{2}\biggr)\,ds+ \frac{4C_{1}\varepsilon_{0} M}{C_{2}\delta} \end{aligned}$$

(4.17)

and consider the family D of closed balls in $X_{0}$ defined by $D_{t}=\{y\in X_{0}, \Vert y \Vert _{X_{0}}\leq R_{t}\}$. It is easy to check the family $D=\{D_{t}\}_{t\in R}$ is a bounded family of pullback absorbing sets in $X_{0}$.

Choose a number δ̃ such that

$$\begin{aligned} \tilde{\delta}< \min\biggl\{ \frac{\eta}{2}, 2 \lambda_{0}^{2} , \frac{\eta-\varepsilon'}{2+5\varepsilon'},\delta,\biggr\} , \end{aligned}$$

(4.18)

where $0<\varepsilon'<\min\{\frac{\sqrt{(9+\lambda_{0}^{2})^{2}+40\eta\lambda_{0}^{2}}-(9+\lambda_{0}^{2})}{20},\eta\}$. Then, reasoning as before, (4.16) is also true if we replace δ by δ̃.

Now, we letting $y_{0}\in D(t_{0}-\tau)$, we deduce that

$$\begin{aligned} \begin{aligned} & \bigl\Vert \Phi(\tau,t_{0}-\tau,y_{0}) \bigr\Vert ^{2}_{X_{0}} \\ &\quad \leq \frac{C_{1}}{C_{2}}\biggl\{ e^{-\tilde{\delta} \tau}C\bigl(R^{2}_{t_{0}-\tau}+R_{t_{0}-\tau} \bigr) +e^{-\tilde{\delta} t_{0}} \int_{-\infty}^{t_{0}} e^{\tilde{\delta} s}\biggl( \frac{2}{\eta} \Vert h \Vert ^{2}+\frac{1}{\lambda_{0}^{2}} \Vert q \Vert ^{2}\biggr)\,ds\\ &\qquad {} +\frac{2\varepsilon_{0} M}{\delta}\bigl(1-e^{-\tilde{\delta}\tau} \bigr)\biggr\} . \end{aligned} \end{aligned}$$

(4.19)

If we set

$$( \tilde{R}_{t})^{2}=2\frac{C_{1}}{C_{2}} e^{-\tilde{\delta} t} \int_{-\infty}^{t}e^{\tilde{\delta} s}\biggl( \frac{2}{\eta} \bigl\Vert h(s) \bigr\Vert ^{2}+ \frac{1}{\lambda_{0}^{2}} \bigl\Vert q(s) \bigr\Vert ^{2}\biggr)\,ds+ \frac{4C_{1}\varepsilon_{0} M}{C_{2}\delta} $$

and

$$\tilde{D}_{t}=\bigl\{ y\in X_{0}, \Vert y \Vert _{X_{0}}\leq \tilde{R}_{t}\bigr\} , $$

then the family $\tilde{D}=\{\tilde{D}_{t}\}_{t\in R}$ satisfies (3.3). The proof is finished. □

5 The pullback attractor in $X_{0}$

In this section, we prove the pullback attractor in $X_{0}$.

Theorem 5.1

Assume that assumptions (H₁)-(H₃) of $g(\cdot)$ hold and that $h(x,t), q(x,t)\in L^{2}_{\mathrm{loc}}(R,H)$ satisfy (1.10) and (1.11) with some δ̃ satisfying $0<\tilde{\delta}<\min\{\frac{\eta}{2},2\lambda_{0}^{2},\frac{\eta-\varepsilon'}{2+5\varepsilon'},\delta\}$ ($0<\varepsilon'<\min\{\frac{\sqrt{(9+\lambda_{0}^{2})^{2}+40\eta\lambda_{0}^{2}}-(9+\lambda_{0}^{2})}{20},\eta\}$). Then there exists a pullback attractor $A=\{A_{t}\}_{t\in R}$ in $X_{0}$ for the nonautonomous dynamical system $(\theta,\Phi)$ defined by (4.1).

Proof

Fix $t_{0}\in R$. Let $y_{i}(t)=(u_{i}(t),v_{i}(t),\theta_{i}(t))$ $(i=1,2)$ be the corresponding weak solution to $y_{0}^{i}=(u_{0}^{i},v_{0}^{i},\theta_{0}^{i})\in \tilde{D}_{t_{0}-\tau}$, where $\tau\geq0$, and let $w(t)=u_{1}(t)-u_{2}(t)$, $\tilde{\theta}(t)=\theta_{1}(t)-\theta_{2}(t)$. Then $(w,\tilde{\theta})$ satisfy

$$\begin{aligned} & w_{tt}+\alpha\triangle^{2}w-\beta\triangle w -\biggl(\sigma\biggl( \int_{\Omega}(\nabla u_{1})^{2}\,dx\biggr) \triangle u_{1} -\sigma\biggl( \int_{\Omega}(\nabla u_{2})^{2}\,dx\biggr) \triangle u_{2}\biggr) \\ &\qquad {} +\gamma\triangle \tilde{\theta} +\triangle g + \eta w_{t} =0, \end{aligned}$$

(5.1)

$$\begin{aligned} & \tilde{\theta}_{t}-\triangle \tilde{\theta}-\gamma\triangle w_{t}=0 \end{aligned}$$

(5.2)

with the initial condition $(w(0),w_{t}(0),\tilde{\theta}(0))=(u_{0}^{1},v_{0}^{1},\theta_{0}^{1})-(u_{0}^{2},v_{0}^{2},\theta_{0}^{2})$, where $\triangle g=g(u_{1})-g(u_{2})$.

Define

$$E_{u}(t)=\frac{1}{2}\bigl( \Vert u_{t} \Vert ^{2}+ \Vert \triangle u \Vert ^{2}+ \Vert \theta \Vert ^{2}\bigr)=\frac{1}{2} \bigl\Vert \phi(t-t_{0}+ \tau, t_{0}-\tau,y_{0}) \bigr\Vert ^{2}_{X_{0}} $$

and

$$F(t)=\frac{1}{2}\bigl( \Vert w_{t} \Vert ^{2}+ \alpha \Vert \triangle w \Vert ^{2}+\beta \Vert \nabla w \Vert ^{2}+\sigma\bigl( \Vert \nabla u_{1} \Vert ^{2} \bigr) \Vert \nabla w \Vert ^{2}+ \Vert \tilde{\theta} \Vert ^{2}\bigr). $$

First, we have

$$\begin{aligned} F(t)&\geq\frac{1}{2}\bigl( \Vert w_{t} \Vert ^{2}+\alpha \Vert \triangle w \Vert ^{2}+ \Vert \tilde{ \theta} \Vert ^{2}\bigr) \\ &\geq\frac{1}{2}C^{1}\bigl( \Vert w_{t} \Vert ^{2}+ \Vert \triangle w \Vert ^{2}+ \Vert \tilde{ \theta} \Vert ^{2}\bigr) \\ &= C^{1}E_{w}(t), \end{aligned}$$

(5.3)

where $C^{1}=\min\{1,\alpha\}$.

Multiplying (5.1) by $e^{\tilde{\delta}t}w_{t}$ and (5.2) by $e^{\tilde{\delta}t}\tilde{\theta}$ and then summing, we obtain

$$\begin{aligned} & \frac{d}{dt}\bigl[e^{\tilde{\delta}t}F(t)\bigr] + e^{\tilde{\delta} t}\bigl(\eta \Vert w_{t} \Vert ^{2}+ \Vert \nabla \tilde{\theta} \Vert ^{2}\bigr) \\ &\quad =\tilde{\delta}e^{\tilde{\delta}t}F(t) \\ &\qquad {}-e^{\tilde{\delta}t}\biggl( \int_{\Omega}\triangle g w_{t}\,dx - \sigma' \bigl( \Vert \nabla u_{1} \Vert ^{2}\bigr) \int_{\Omega}\nabla u_{1}\nabla u_{1t}\,dx \Vert \nabla w \Vert ^{2} - \int_{\Omega}\triangle\sigma w_{t}\,dx\biggr), \end{aligned}$$

(5.4)

where $\triangle\sigma=\sigma( \Vert \nabla u_{1} \Vert ^{2})-\sigma( \Vert \nabla u_{2} \Vert ^{2})$, and δ̃ satisfies (4.18).

Integrating (5.4) from s to $t_{0}$, we have

$$\begin{aligned} &e^{\tilde{\delta}t_{0}}F(t_{0})-e^{\tilde{\delta}s}F(s)+ \int_{s}^{t_{0}}e^{\tilde{\delta}\xi}\bigl(\eta \Vert w_{t} \Vert ^{2}+ \Vert \nabla \tilde{\theta} \Vert ^{2}\bigr)\,d\xi \\ &\quad =\tilde{\delta} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi}F(\xi)\,d\xi - \int_{s}^{t_{0}}e^{\tilde{\delta}\xi}\biggl( \int_{\Omega}\triangle g w_{t}\,dx\biggr)\,d\xi \\ &\qquad {}+ \int_{s}^{t_{0}}e^{\tilde{\delta}\xi}\biggl( \sigma'\bigl( \Vert \nabla u_{1} \Vert ^{2} \bigr) \int_{\Omega}\nabla u_{1}\nabla u_{1t}\,dx \Vert \nabla w \Vert ^{2} + \int_{\Omega}\triangle\sigma w_{t}\,dx\biggr)\,d\xi . \end{aligned}$$

(5.5)

Integrating (5.5) from $t_{0}-\tau$ to $t_{0}$ with respect to s, we obtain

$$\begin{aligned} &\tau e^{\tilde{\delta}t_{0}}F(t_{0})- \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}s}F(s)\,ds + \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi}\bigl(\eta \Vert w_{t} \Vert ^{2}+ \Vert \nabla \tilde{\theta} \Vert ^{2}\bigr)\,d\xi \,ds \\ &\quad =\tilde{\delta} \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi}F(\xi)\,d\xi \,ds - \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi}\biggl( \int_{\Omega}\triangle g w_{t}\,dx\biggr)\,d\xi \,ds \\ &\qquad {}+ \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi}\biggl( \sigma'\bigl( \Vert \nabla u_{1} \Vert ^{2} \bigr) \int_{\Omega}\nabla u_{1}\nabla u_{1t}\,dx \Vert \nabla w \Vert ^{2} + \int_{\Omega}\triangle\sigma w_{t}\,dx\biggr)\,d\xi \,ds. \end{aligned}$$

(5.6)

Similarly, multiplying (5.1) by $e^{\tilde{\delta}t}w$, we have

$$\begin{aligned} &\frac{d}{dt}\bigl[e^{\tilde{\delta}t}(w_{t},w) \bigr]+e^{\tilde{\delta}t}\bigl(\alpha \Vert \triangle w \Vert ^{2}+ \beta \Vert \nabla w \Vert ^{2}+\sigma\bigl( \Vert \nabla u_{1} \Vert ^{2}\bigr) \Vert \nabla w \Vert ^{2}\bigr) \\ &\quad =(\tilde{\delta}-\eta)e^{\tilde{\delta}t}(w_{t},w)+e^{\tilde{\delta}t} \Vert w_{t} \Vert ^{2} \\ &\qquad {}-e^{\tilde{\delta}t}\biggl( \int_{\Omega}\triangle g w\,dx -\gamma \int_{\Omega}\triangle \tilde{\theta} w\,dx - \int_{\Omega}\triangle\sigma \triangle u_{2} w \,dx \biggr). \end{aligned}$$

(5.7)

First, integrating (5.7) over $[s,t_{0}]$, we get that

$$\begin{aligned} \begin{aligned}[b] & \int_{s}^{t_{0}}e^{\tilde{\delta}\xi}\bigl(\alpha \Vert \triangle w \Vert ^{2}+\beta \Vert \nabla w \Vert ^{2}+ \sigma\bigl( \Vert \nabla u_{1} \Vert ^{2}\bigr) \Vert \nabla w \Vert ^{2}\bigr)\,d\xi + e^{\tilde{\delta}t_{0}} \bigl(w_{t}(t_{0}),w(t_{0})\bigr) \\ &\quad =e^{\tilde{\delta}s}\bigl(w_{t}(s),w(s)\bigr)+(\tilde{\delta}-\eta) \int_{s}^{t_{0}}e^{\tilde{\delta}\xi}(w_{t},w)\,d \xi + \int_{s}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w_{t} \Vert ^{2}\,d\xi \\ &\qquad {}- \int_{s}^{t_{0}} \int_{\Omega}e^{\tilde{\delta}\xi}\triangle g w\,dx\,d\xi -\gamma \int_{s}^{t_{0}} \int_{\Omega}e^{\tilde{\delta}\xi}\triangle \tilde{\theta} w\,dx\,d\xi\\ &\qquad {} - \int_{s}^{t_{0}} \int_{\Omega}e^{\tilde{\delta}\xi}\triangle\sigma \triangle u_{2} w \,dx\,d\xi . \end{aligned} \end{aligned}$$

(5.8)

Then integrating (5.8) over $[t_{0}-\tau,t_{0}]$ with respect to s, we obtain

$$\begin{aligned} &\tilde{\delta} \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi}\bigl(\alpha \Vert \triangle w \Vert ^{2}+\beta \Vert \nabla w \Vert ^{2}+ \sigma\bigl( \Vert \nabla u_{1} \Vert ^{2}\bigr) \Vert \nabla w \Vert ^{2}\bigr)\,d\xi \,ds \\ &\quad =-\tilde{\delta}\tau e^{\tilde{\delta}t_{0}}\bigl(w_{t}(t_{0}),w(t_{0}) \bigr) + \tilde{\delta} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}s}\bigl(w_{t}(s),w(s) \bigr)\,ds \\ &\qquad {}+\tilde{\delta}(\tilde{\delta}-\eta) \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi}(w_{t},w)\,d \xi \,ds +\tilde{\delta} \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w_{t} \Vert ^{2}\,d\xi \,ds \\ &\qquad {}-\tilde{\delta} \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}} \int_{\Omega}e^{\tilde{\delta}\xi}\triangle g w\,dx\,d\xi \,ds -\gamma \tilde{\delta} \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}} \int_{\Omega}e^{\tilde{\delta}\xi}\triangle \tilde{\theta} w\,dx\,d\xi \,ds \\ &\qquad {}-\tilde{\delta} \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}} \int_{\Omega}e^{\tilde{\delta}\xi}\triangle\sigma \triangle u_{2} w \,dx\,d\xi \,ds. \end{aligned}$$

(5.9)

Substituting (5.9) into (5.6), we deduce that

$$\begin{aligned} &\tau e^{\tilde{\delta}t_{0}}F(t_{0})- \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}s}F(s)\,ds + \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi}\bigl(\eta \Vert w_{t} \Vert ^{2}+ \Vert \nabla \tilde{\theta} \Vert ^{2}\bigr)\,d\xi \,ds \\ &\quad =\tilde{\delta} \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi} \frac{1}{2} \bigl( \Vert w_{t} \Vert ^{2}+\alpha \Vert \triangle w \Vert ^{2}+\beta \Vert \nabla w \Vert ^{2}+\sigma\bigl( \Vert \nabla u_{1} \Vert ^{2}\bigr) \Vert \nabla w \Vert ^{2}+ \Vert \tilde{\theta} \Vert ^{2}\bigr) \,d\xi \,ds \\ &\qquad {}- \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi}\biggl( \int_{\Omega}\triangle g w_{t}\,dx - \sigma' \bigl( \Vert \nabla u_{1} \Vert ^{2}\bigr) \int_{\Omega}\nabla u_{1}\nabla u_{1t}\,dx \Vert \nabla w \Vert ^{2} \\ &\qquad {}- \int_{\Omega}\triangle\sigma w_{t}\,dx\biggr)\,d\xi \,ds \\ &\quad =\tilde{\delta} \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi} \frac{1}{2} \bigl( \Vert w_{t} \Vert ^{2}+ \Vert \tilde{\theta} \Vert ^{2}\bigr)\,d\xi \,ds -\frac{1}{2}\tilde{\delta}\tau e^{\tilde{\delta}t_{0}}\bigl(w_{t}(t_{0}),w(t_{0})\bigr) \\ &\qquad {}+ \frac{1}{2}\tilde{\delta} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}s}\bigl(w_{t}(s),w(s) \bigr)\,ds +\frac{1}{2}\tilde{\delta}(\tilde{\delta}-\eta) \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi}(w_{t},w)\,d \xi \,ds \\ &\qquad {}+\frac{1}{2}\tilde{\delta} \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w_{t} \Vert ^{2}\,d\xi \,ds -\frac{1}{2}\tilde{\delta} \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}} \int_{\Omega}e^{\tilde{\delta}\xi}\triangle g w\,dx\,d\xi \,ds \\ &\qquad {}-\frac{1}{2}\gamma\tilde{\delta} \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}} \int_{\Omega}e^{\tilde{\delta}\xi}\triangle \tilde{\theta} w\,dx\,d\xi \,ds -\frac{1}{2}\tilde{\delta} \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}} \int_{\Omega}e^{\tilde{\delta}\xi}\triangle\sigma \triangle u_{2} w \,dx\,d\xi \,ds \\ &\qquad {}- \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi}\biggl( \int_{\Omega}\triangle g w_{t}\,dx - \sigma' \bigl( \Vert \nabla u_{1} \Vert ^{2}\bigr) \int_{\Omega}\nabla u_{1}\nabla u_{1t}\,dx \Vert \nabla w \Vert ^{2} \\ &\qquad {}- \int_{\Omega}\triangle\sigma w_{t}\,dx\biggr)\,d\xi \,ds. \end{aligned}$$

(5.10)

Second, integrating (5.7) from $t_{0}-\tau$ to $t_{0}$, we have

$$\begin{aligned} & \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi}\bigl( \Vert w_{t} \Vert ^{2}+\alpha \Vert \triangle w \Vert ^{2}+\beta \Vert \nabla w \Vert ^{2}+\sigma\bigl( \Vert \nabla u_{1} \Vert ^{2}\bigr) \Vert \nabla w \Vert ^{2}+ \Vert \tilde{\theta} \Vert ^{2}\bigr)\,d\xi \\ &\quad =e^{\tilde{\delta}(t_{0}-\tau)}\bigl(w_{t}(t_{0}-\tau),w(t_{0}- \tau)\bigr)-e^{\tilde{\delta}t_{0}}\bigl(w_{t}(t_{0}),w(t_{0}) \bigr) \\ &\qquad {}+(\tilde{\delta}-\eta) \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi}(w_{t},w)\,d \xi + \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w_{t} \Vert ^{2}\,d\xi \\ &\qquad {}- \int_{t_{0}-\tau}^{t_{0}} \int_{\Omega}e^{\tilde{\delta}\xi}\triangle g w\,dx \,d\xi -\gamma \int_{t_{0}-\tau}^{t_{0}} \int_{\Omega}e^{\tilde{\delta}\xi}\triangle \tilde{\theta} w\,dx\,d\xi \\ &\qquad {} - \int_{t_{0}-\tau}^{t_{0}} \int_{\Omega}e^{\tilde{\delta}\xi}\triangle\sigma \triangle u_{2} w \,dx\,d\xi + \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi}\bigl( \Vert w_{t} \Vert ^{2}+ \Vert \tilde{\theta} \Vert ^{2}\bigr)\,d\xi . \end{aligned}$$

(5.11)

Substituting (5.11) into (5.10) and noting that $\tilde{\delta}<\frac{\eta}{2}$, we have

$$\begin{aligned} &\tau e^{\tilde{\delta}t_{0}}F(t_{0})+ \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}s}F(s)\,ds + \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi} \Vert \nabla \tilde{ \theta} \Vert ^{2}\,d\xi \,ds \\ &\quad \leq\frac{\tilde{\delta}}{2} \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi} \Vert \tilde{\theta} \Vert ^{2}\,d\xi \,ds -\biggl(\frac{1}{2}\tilde{\delta}\tau+1 \biggr) e^{\tilde{\delta}t_{0}}\bigl(w_{t}(t_{0}),w(t_{0}) \bigr) \\ &\qquad {}+ \frac{1}{2}\tilde{\delta} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}s}\bigl(w_{t}(s),w(s) \bigr)\,ds +\frac{1}{2}\tilde{\delta}(\tilde{\delta}-\eta) \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi}(w_{t},w)\,d \xi \,ds \\ &\qquad {}-\frac{1}{2}\tilde{\delta} \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}} \int_{\Omega}e^{\tilde{\delta}\xi}\triangle g w\,dx\,d\xi \,ds - \frac{1}{2}\gamma\tilde{\delta} \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}} \int_{\Omega}e^{\tilde{\delta}\xi}\triangle \tilde{\theta} w\,dx\,d\xi \,ds \\ &\qquad {}-\frac{1}{2}\tilde{\delta} \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}} \int_{\Omega}e^{\tilde{\delta}\xi}\triangle\sigma\triangle u_{2} w \,dx\,d\xi \,ds + \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi}\biggl( \int_{\Omega}\triangle g w_{t}\,dx\biggr)\,d\xi \,ds \\ &\qquad {}- \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi}\biggl( \sigma'\bigl( \Vert \nabla u_{1} \Vert ^{2} \bigr) \int_{\Omega}\nabla u_{1}\nabla u_{1t}\,dx \Vert \nabla w \Vert ^{2} + \int_{\Omega}\triangle\sigma w_{t}\,dx\biggr)\,d\xi \,ds \\ &\qquad {}+e^{\tilde{\delta}(t_{0}-\tau)}\bigl(w_{t}(t_{0}-\tau),w(t_{0}- \tau)\bigr) +(\tilde{\delta}-\eta) \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi}(w_{t},w)\,d \xi + \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w_{t} \Vert ^{2}\,d\xi \\ &\qquad {}- \int_{t_{0}-\tau}^{t_{0}} \int_{\Omega}e^{\tilde{\delta}\xi}\triangle g w\,dx \,d\xi -\gamma \int_{t_{0}-\tau}^{t_{0}} \int_{\Omega}e^{\tilde{\delta}\xi}\triangle \tilde{\theta} w\,dx\,d\xi \\ &\qquad {} - \int_{t_{0}-\tau}^{t_{0}} \int_{\Omega}e^{\tilde{\delta}\xi}\triangle\sigma \triangle u_{2} w \,dx\,d\xi + \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi}\bigl( \Vert w_{t} \Vert ^{2}+ \Vert \tilde{\theta} \Vert ^{2}\bigr)\,d\xi . \end{aligned}$$

(5.12)

• (I) Using the Schwarz and Young inequalities, for $t_{0}-\tau\leq s< t_{0}$, we have

  $$\begin{aligned} &\frac{1}{2}\tilde{\delta} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}s}\bigl(w_{t}(s),w(s) \bigr)\,ds \leq\frac{\tilde{\delta}^{2}}{8\varepsilon'} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}s} \bigl\Vert w(s) \bigr\Vert ^{2}\,ds +\varepsilon' \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}s} \bigl\Vert w_{t}(s) \bigr\Vert ^{2}\,ds, \end{aligned}$$

  (5.13)
  $$\begin{aligned} &\begin{aligned}[b] &\frac{1}{2}\tilde{\delta}(\tilde{ \delta}-\eta) \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi}(w_{t},w)\,d \xi \,ds \\ &\quad \leq \frac{1}{2}\tau\tilde{\delta}(\tilde{\delta}-\eta) \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w_{t} \Vert \Vert w \Vert \,d\xi \\ &\quad \leq \frac{[\frac{1}{2}\tau\tilde{\delta}(\tilde{\delta}-\eta)]^{2}}{4\varepsilon'} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w \Vert ^{2}\,d\xi +\varepsilon' \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w_{t} \Vert ^{2}\,d\xi , \end{aligned} \end{aligned}$$

  (5.14)
  $$\begin{aligned} & \begin{aligned}[b] & {-}\frac{1}{2}\gamma\tilde{\delta} \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}} \int_{\Omega}e^{\tilde{\delta}\xi}\triangle \tilde{\theta} w\,dx\,d\xi \,ds\\ &\quad \leq \frac{\varepsilon'}{\lambda_{0}^{2}} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert \nabla\tilde{ \theta} \Vert ^{2}\,d\xi +\frac{(\tau \frac{1}{2}\gamma\tilde{\delta})^{2}}{4\varepsilon'} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert \nabla w \Vert ^{2}\,d\xi , \end{aligned} \end{aligned}$$

  (5.15)
  $$\begin{aligned} &(\tilde{\delta}-\eta) \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi}(w_{t},w)\,d \xi \leq\frac{(\tilde{\delta}-\eta)^{2}}{8\varepsilon'} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\sigma}\xi} \Vert w \Vert ^{2}\,d\xi +2\varepsilon' \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\sigma}\xi} \Vert w_{t} \Vert ^{2}\,d\xi , \end{aligned}$$

  (5.16)

  and

  $$\begin{aligned} -\gamma \int_{t_{0}-\tau}^{t_{0}} \int_{\Omega}e^{\tilde{\delta}\xi}\triangle \tilde{\theta} w\,dx\,d\xi \leq \frac{\varepsilon'}{\lambda_{0}^{2}} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert \nabla \tilde{ \theta} \Vert ^{2} \,d\xi +\frac{\lambda_{0}^{2}\gamma^{2}}{4\varepsilon'} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert \nabla w \Vert ^{2}\,d\xi , \end{aligned}$$

  (5.17)

  where $0<\varepsilon'<\min\{\frac{\sqrt{(9+\lambda_{0}^{2})^{2}+40\eta\lambda_{0}^{2}}-(9+\lambda_{0}^{2})}{20},\eta\}$.

• (II) By assumption (1.6) on $g(\cdot)$, the Hölder inequality, and the embedding theorem combined with (4.19), for $t_{0}-\tau\leq s< t_{0}$, we have

  $$\begin{aligned} &\frac{1}{2}\tilde{\delta} \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}} \int_{\Omega}e^{\tilde{\delta}\xi}\triangle g w\,dx\,d\xi \,ds \\ &\quad \leq\frac{1}{2}\tilde{\delta}\tau\biggl( \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \int_{\Omega} \bigl\vert g(u_{2})-g(u_{1}) \bigr\vert ^{2}\,dx\,d\xi \biggr)^{\frac{1}{2}}\biggl( \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w \Vert ^{2}\,d\xi \biggr)^{\frac{1}{2}} \\ &\quad \leq \frac{1}{2}\tilde{\delta}C\tau\biggl( \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \int_{\Omega}\bigl(1+ \vert u_{1} \vert ^{2\rho-2}+ \vert u_{2} \vert ^{2\rho-2}\bigr) \vert w \vert ^{2}\,dx\,d\xi \biggr)^{\frac{1}{2}} \\ &\qquad {}\times\biggl( \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w \Vert ^{2}\,d\xi \biggr)^{\frac{1}{2}} \\ &\quad \leq \frac{1}{2}\tilde{\delta}C\tau\biggl( \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \int_{\Omega}\bigl( \vert u_{1} \vert ^{2}+ \vert u_{2} \vert ^{2}+ \vert u_{1} \vert ^{2\rho} + \vert u_{2} \vert ^{2\rho}\bigr)\,dx\,d\xi \biggr)^{\frac{1}{2}} \\ &\qquad {}\times\biggl( \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w \Vert ^{2}\,d\xi \biggr)^{\frac{1}{2}} \\ &\quad \leq \frac{1}{2}\tilde{\delta}C\tau\biggl( \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi}\bigl( \Vert \nabla u_{1} \Vert ^{2}+ \Vert \nabla u_{2} \Vert ^{2} + \Vert \nabla u_{1} \Vert ^{2\rho}+ \Vert \nabla u_{2} \Vert ^{2\rho}\bigr)\,d\xi \biggr)^{\frac{1}{2}} \\ &\qquad {}\times \biggl( \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w \Vert ^{2}\,d\xi \biggr)^{\frac{1}{2}} \\ &\quad \leq C_{t_{0},\tau}\biggl( \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w \Vert ^{2}\,d\xi \biggr)^{\frac{1}{2}}, \end{aligned}$$

  (5.18)

  and similarly we also have

  $$\begin{aligned} - \int_{t_{0}-\tau}^{t_{0}} \int_{\Omega}e^{\tilde{\delta}\xi}\triangle g w\,dx \,d\xi \leq C_{t_{0},\tau}\biggl( \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\sigma}\xi} \Vert w \Vert ^{2}\,d\xi \biggr)^{\frac{1}{2}}. \end{aligned}$$

  (5.19)

• (III) By the value theorem, (4.18), the continuity of $\sigma(\cdot)$, and the Schwarz inequality, for $t_{0}-\tau\leq s< t_{0}$, we obtain that

  $$\begin{aligned} &-\frac{1}{2}\tilde{\delta} \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}} \int_{\Omega}e^{\tilde{\delta}\xi}\triangle\sigma\triangle u_{2} w \,dx\,d\xi \,ds \\ &\quad \leq\frac{1}{2}\tilde{\delta}\tau \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi}\sigma'( \xi_{1}) \bigl( \Vert \triangle u_{1} \Vert ^{2}- \Vert \triangle u_{2} \Vert ^{2}\bigr) \Vert \triangle u_{2} \Vert \Vert w \Vert \,d\xi \\ &\quad \leq C_{t_{0},\tau} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w \Vert ^{2} \,d\xi , \end{aligned}$$

  (5.20)

  where $\xi_{1}$ is between $\Vert \nabla u_{1} \Vert ^{2}$ and $\Vert \nabla u_{2} \Vert ^{2}$. Similarly, we have

  $$\begin{aligned} & \int_{t_{0}-\tau}^{t_{0}} \int_{\Omega}e^{\tilde{\delta}\xi}\triangle\sigma \triangle u_{2} w \,dx\,d\xi \leq C_{t_{0},\tau} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w \Vert ^{2} \,d\xi , \end{aligned}$$

  (5.21)
  $$\begin{aligned} &{-} \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi} \sigma' \bigl( \Vert \nabla u_{1} \Vert ^{2}\bigr) \int_{\Omega}\nabla u_{1}\nabla u_{1t}\,dx \Vert \nabla w \Vert ^{2}\,d\xi \,ds \\ &\quad \leq C \tau C_{t_{0},\tau} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert \nabla w \Vert ^{2}\,d\xi , \end{aligned}$$

  (5.22)

  and

  $$\begin{aligned} & {-} \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi} \int_{\Omega}\triangle\sigma w_{t}\,dx\,d\xi \,ds \\ &\quad \leq \tau C_{t_{0},\tau} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w \Vert \Vert w_{t} \Vert \,d\xi \\ &\quad \leq \frac{(\tau C_{t_{0},\tau})^{2}}{4\varepsilon'} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w \Vert ^{2}\,d\xi +\varepsilon' \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w_{t} \Vert ^{2}\,d\xi . \end{aligned}$$

  (5.23)

• (IV) Since $\tilde{\delta}\leq2\lambda_{0}^{2}$, we have

  $$\begin{aligned} \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi} \Vert \nabla \tilde{ \theta} \Vert ^{2}\,d\xi \,ds &\geq\lambda_{0}^{2} \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi} \Vert \tilde{\theta} \Vert ^{2}\,d\xi \\ &\geq\frac{\tilde{\delta}}{2} \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi} \Vert \tilde{\theta} \Vert ^{2}\,d\xi \,ds. \end{aligned}$$

  (5.24)

  So, substituting (5.13)-(5.23) into (5.12), by (5.24) we obtain

  $$\begin{aligned} &\tau e^{\tilde{\delta}t_{0}}F(t_{0})+ \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}s}F(s)\,ds \\ &\quad \leq \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert \tilde{\theta} \Vert ^{2}\,d\xi -\biggl(\frac{1}{2}\tilde{\delta}\tau+1 \biggr) e^{\tilde{\delta}t_{0}}\bigl(w_{t}(t_{0}),w(t_{0}) \bigr) \\ &\qquad {}+\biggl(\frac{\tilde{\delta}^{2}}{8\varepsilon'}+\frac{[\frac{1}{2}\tau\tilde{\delta} (\tilde{\delta}-\eta)]^{2}}{4\varepsilon'}+\frac{(\tau C_{t_{0},\tau})^{2}}{4\varepsilon'}+ \frac{(\tilde{\delta}-\eta)^{2}}{8\varepsilon'}+2C_{t_{0},\tau}\biggr) \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w \Vert ^{2}\,d\xi \\ &\qquad {}+\bigl(2+5\varepsilon'\bigr) \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w_{t} \Vert ^{2}\,d\xi +2C_{t_{0},\tau}\biggl( \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w \Vert ^{2}\,d\xi \biggr)^{\frac{1}{2}} \\ & \qquad {}+\frac{2\varepsilon'}{\lambda_{0}^{2}} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert \nabla\tilde{ \theta} \Vert ^{2}\,d\xi \\ & \qquad {} +\biggl(\frac{(\tau \frac{1}{2}\gamma\tilde{\delta})^{2}}{4\varepsilon'}+ \frac{\lambda_{0}^{2}\gamma^{2}}{4\varepsilon'}+C \tau C_{t_{0},\tau}+\tau C_{t_{0},\tau}\biggr) \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert \nabla w \Vert ^{2}\,d\xi \\ &\qquad {}+ \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi} \int_{\Omega}\triangle g w_{t}\,dx\,d\xi \,ds+e^{\tilde{\delta}(t_{0}-\tau)}\bigl(w_{t}(t_{0}-\tau),w(t_{0}- \tau)\bigr). \end{aligned}$$

  (5.25)

On the other hand, integrating (5.4) over $[t_{0}-\tau,t_{0}]$, we get that

$$\begin{aligned} & e^{\tilde{\delta}t_{0}}F(t_{0})-e^{\tilde{\delta}(t_{0}-\tau)}F(t_{0}- \tau) + \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta} \xi}\bigl(\eta \Vert w_{t} \Vert ^{2}+ \Vert \nabla \tilde{\theta} \Vert ^{2}\bigr)\,d\xi \\ &\quad =\tilde{\delta} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi}F(\xi)\,d\xi - \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \int_{\Omega}\triangle g w_{t}\,dx\,d\xi \\ &\qquad {}- \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \biggl( \sigma'\bigl( \Vert \nabla u \Vert ^{2}\bigr) \int_{\Omega}\nabla u_{1}\nabla u_{1t}\,dx \Vert \nabla w \Vert ^{2} - \int_{\Omega}\triangle\sigma w_{t}\,dx\biggr)\,d\xi . \end{aligned}$$

(5.26)

By the continuity of $\sigma'(\cdot)$ combined with (4.19) we get

$$\begin{aligned} - \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi}\sigma' \bigl( \Vert \nabla u_{1} \Vert ^{2}\bigr) \int_{\Omega}\nabla u_{1}\nabla u_{1t}\,dx \Vert \nabla w \Vert ^{2}\,d\xi \leq C_{t_{0},\tau} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert \nabla w \Vert ^{2}\,d\xi . \end{aligned}$$

(5.27)

By the value theorem and (4.19) we have

$$\begin{aligned} \begin{aligned}[b] - \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \int_{\Omega}\triangle\sigma w_{t}\,dx\,d\xi &\leq C_{t_{0},\tau} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w \Vert \Vert w_{t} \Vert \,d\xi \\ &\leq \frac{C_{t_{0},\tau}^{2}}{4\varepsilon'} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w \Vert ^{2}\,d\xi +\varepsilon' \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w_{t} \Vert ^{2}\,d\xi . \end{aligned} \end{aligned}$$

(5.28)

From (5.26) combined with (5.27)-(5.28) we have

$$\begin{aligned} & \bigl(2+5\varepsilon'\bigr) \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta} \xi} \Vert w_{t} \Vert ^{2}\,d\xi \\ &\quad \leq \frac{(2+5\varepsilon')}{\eta-\varepsilon'}e^{\tilde{\delta}(t_{0}-\tau)}F(t_{0}-\tau) + \frac{(2+5\varepsilon')\tilde{\delta}}{\eta-\varepsilon'} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi}F(\xi)\,d\xi \\ &\qquad {}-\frac{(2+5\varepsilon')}{\eta-\varepsilon'} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \int_{\Omega}\triangle g w_{t}\,dx\,d\xi + \frac{(2+5\varepsilon')}{\eta-\varepsilon'}C_{t_{0},\tau} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert \nabla w \Vert ^{2} \,d\xi \\ &\qquad {} +\frac{(2+5\varepsilon')}{\eta-\varepsilon'}\frac{C^{2}_{t_{0},\tau}}{4\varepsilon'} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w \Vert ^{2} \,d\xi -\frac{(2+5\varepsilon')}{\eta-\varepsilon'} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert \nabla \tilde{ \theta} \Vert ^{2}\,d\xi . \end{aligned}$$

(5.29)

Substituting (5.29) into (5.25), we obtain

$$\begin{aligned} &\tau e^{\tilde{\delta}t_{0}}F(t_{0})+ \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}s}F(s)\,ds \\ &\quad \leq \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert \tilde{\theta} \Vert ^{2}\,d\xi -\biggl(\frac{1}{2}\tilde{\delta}\tau+1 \biggr) e^{\tilde{\delta}t_{0}}\bigl(w_{t}(t_{0}),w(t_{0}) \bigr) + C^{2}_{t_{0},\tau} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w \Vert ^{2}\,d\xi \\ &\qquad {}+\frac{(2+5\varepsilon')}{\eta-\varepsilon'}e^{\tilde{\delta}(t_{0}-\tau)}F(t_{0}-\tau) + \frac{(2+5\varepsilon')\tilde{\delta}}{\eta-\varepsilon'} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi}F(\xi)\,d\xi \\ &\qquad {}-\frac{(2+5\varepsilon')}{\eta-\varepsilon'} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \int_{\Omega}\triangle g w_{t}\,dx\,d\xi - \frac{(2+5\varepsilon')}{\eta-\varepsilon'} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert \nabla \tilde{ \theta} \Vert ^{2}\,d\xi \\ &\qquad {}+2C_{t_{0},\tau}\biggl( \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w \Vert ^{2}\,d\xi \biggr)^{\frac{1}{2}}+\frac{2\varepsilon'}{\lambda_{0}^{2}} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert \nabla\tilde{ \theta} \Vert ^{2}\,d\xi +C^{3}_{t_{0},\tau} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert \nabla w \Vert ^{2}\,d\xi \\ &\qquad {}+ \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi} \int_{\Omega}\triangle g w_{t}\,dx\,d\xi \,ds +e^{\tilde{\delta}(t_{0}-\tau)}\bigl(w_{t}(t_{0}-\tau),w(t_{0}- \tau)\bigr), \end{aligned}$$

(5.30)

where $C^{2}_{t_{0},\tau}=\frac{\tilde{\delta}^{2}}{8\varepsilon'}+\frac{[\frac{1}{2}\tau\tilde{\delta} (\tilde{\delta}-\eta)]^{2}}{4\varepsilon'}+\frac{(\tau C_{t_{0},\tau})^{2}}{4\varepsilon'}+\frac{(\tilde{\delta}-\eta)^{2}}{8\varepsilon'}+2C_{t_{0},\tau}+\frac{(2+5\varepsilon')}{\eta-\varepsilon'}\frac{C^{2}_{t_{0},\tau}}{4\varepsilon'}$ and $C^{3}_{t_{0},\tau}=\frac{(\tau \frac{1}{2}\gamma\tilde{\delta})^{2}}{4\varepsilon'}+\frac{\lambda_{0}^{2}\gamma^{2}}{4\varepsilon'}+C \tau C_{t_{0},\tau}+\tau C_{t_{0},\tau}+\frac{(2+5\varepsilon')}{\eta-\varepsilon'}C_{t_{0},\tau}$.

Since $0<\varepsilon'<\min\{\frac{\sqrt{(9+\lambda_{0}^{2})^{2}+40\eta\lambda_{0}^{2}}-(9+\lambda_{0}^{2})}{20},\eta\}$, we have

$$-\frac{(2+5\varepsilon')}{\eta-\varepsilon'} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert \nabla \tilde{ \theta} \Vert ^{2}\,d\xi +\frac{2\varepsilon'}{\lambda_{0}^{2}} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert \nabla\tilde{ \theta} \Vert ^{2}\,d\xi + \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert \tilde{\theta} \Vert ^{2}\,d\xi \leq0. $$

So from (5.30) we obtain

$$\begin{aligned} \begin{aligned}[b] &\tau e^{\tilde{\delta}t_{0}}F(t_{0})+ \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}s}F(s)\,ds \\ &\quad \leq -\biggl(\frac{1}{2}\tilde{\delta}\tau+1\biggr) e^{\tilde{\delta}t_{0}} \bigl(w_{t}(t_{0}),w(t_{0})\bigr) +C^{2}_{t_{0},\tau} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w \Vert ^{2}\,d\xi \\ &\qquad {}+\frac{(2+5\varepsilon')}{\eta-\varepsilon'}e^{\tilde{\delta}(t_{0}-\tau)}F(t_{0}-\tau) + \frac{(2+5\varepsilon')\tilde{\delta}}{\eta-\varepsilon'} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi}F(\xi)\,d\xi \\ &\qquad {}-\frac{2(2+5\varepsilon')}{\eta-\varepsilon'} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \int_{\Omega}\triangle g w_{t}\,dx\,d\xi \\ &\qquad {}+2C_{t_{0},\tau}\biggl( \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w \Vert ^{2}\,d\xi \biggr)^{\frac{1}{2}}+C^{3}_{t_{0},\tau} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert \nabla w \Vert ^{2}\,d\xi \\ &\qquad {}+ \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi} \int_{\Omega}\triangle g w_{t}\,dx\,d\xi \,ds +e^{\tilde{\delta}(t_{0}-\tau)}\bigl(w_{t}(t_{0}-\tau),w(t_{0}- \tau)\bigr). \end{aligned} \end{aligned}$$

(5.31)

Since $\tilde{\delta}\leq\frac{\eta-\varepsilon'}{2+5\varepsilon'}$, we have

$$\begin{aligned} \begin{aligned}[b] \tau e^{\tilde{\delta}t_{0}}F(t_{0}) \leq{} &{-}\biggl(\frac{1}{2}\tilde{\delta}\tau+1\biggr) e^{\tilde{\delta}t_{0}} \bigl(w_{t}(t_{0}),w(t_{0})\bigr) +C^{2}_{t_{0},\tau} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w \Vert ^{2}\,d\xi \\ &{}+\frac{(2+5\varepsilon')}{\eta-\varepsilon'}e^{\tilde{\delta}(t_{0}-\tau)}F(t_{0}-\tau) - \frac{(2+5\varepsilon')}{\eta-\varepsilon'} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \int_{\Omega}\triangle g w_{t}\,dx\,d\xi \\ &{}+2C_{t_{0},\tau}\biggl( \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w \Vert ^{2}\,d\xi \biggr)^{\frac{1}{2}}+C^{3}_{t_{0},\tau} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert \nabla w \Vert ^{2}\,d\xi \\ &{}+ \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi} \int_{\Omega}\triangle g w_{t}\,dx\,d\xi \,ds +e^{\tilde{\delta}(t_{0}-\tau)}\bigl(w_{t}(t_{0}-\tau),w(t_{0}- \tau)\bigr). \end{aligned} \end{aligned}$$

(5.32)

By (5.3) we have

$$\begin{aligned} \begin{aligned}[b] E_{w}(t_{0}) \leq{}&{-} \frac{1}{C^{1}}\biggl(\frac{1}{2}\tilde{\delta}+\frac{1}{\tau} \biggr) \bigl(w_{t}(t_{0}),w(t_{0})\bigr) + \frac{1}{C^{1}\tau} C^{2}_{t_{0},\tau}e^{-\tilde{\delta}t_{0}} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w \Vert ^{2}\,d\xi \\ &{}+\frac{(2+5\varepsilon')}{C^{1}(\eta-\varepsilon')\tau}e^{\tilde{\delta}(-\tau)}F(t_{0}-\tau) - \frac{(2+5\varepsilon')}{C^{1}(\eta-\varepsilon')\tau}e^{-\tilde{\delta}t_{0}} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \int_{\Omega}\triangle g w_{t}\,dx\,d\xi \\ &{} +2C_{t_{0},\tau}\biggl( \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w \Vert ^{2}\,d\xi \biggr)^{\frac{1}{2}}+\frac{1}{C^{1}\tau}C^{3}_{t_{0},\tau} e^{-\tilde{\delta}t_{0}} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert \nabla w \Vert ^{2}\,d\xi \\ &{}+\frac{1}{C^{1}\tau}e^{-\tilde{\delta}t_{0}} \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi} \int_{\Omega}\triangle g w_{t}\,dx\,d\xi \,ds + \frac{1}{C^{1}\tau}e^{-\tilde{\delta}\tau}\bigl(w_{t}(t_{0}- \tau),w(t_{0}-\tau)\bigr). \end{aligned} \end{aligned}$$

(5.33)

We set

$$\begin{aligned} \begin{aligned}[b] &\phi_{t_{0},\tau}\bigl(\bigl(u_{0}^{1},v_{0}^{1}, \theta_{0}^{1}\bigr),\bigl(u_{0}^{2},v_{0}^{2}, \theta_{0}^{2}\bigr)\bigr) \\ &\quad =-\frac{1}{C^{1}}\biggl(\frac{1}{2}\tilde{\delta}+ \frac{1}{\tau}\biggr) \bigl(w_{t}(t_{0}),w(t_{0}) \bigr) +\frac{1}{C^{1}\tau} e^{-\tilde{\delta}t_{0}}C^{2}_{t_{0},\tau} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w \Vert ^{2}\,d\xi \\ &\qquad {}-\frac{(2+5\varepsilon')}{C^{1}\lambda^{2}\tau}e^{-\tilde{\delta}t_{0}} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \int_{\Omega}\triangle g w_{t}\,dx\,d\xi +2C_{t_{0},\tau}\biggl( \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w \Vert ^{2}\,d\xi \biggr)^{\frac{1}{2}} \\ &\qquad {}+\frac{1}{C^{1}\tau}C^{3}_{t_{0},\tau} e^{-\tilde{\delta}t_{0}} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert \nabla w \Vert ^{2}\,d\xi +\frac{1}{C^{1}\tau}e^{-\tilde{\delta}t_{0}} \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi} \int_{\Omega}\triangle g w_{t}\,dx\,d\xi \,ds. \end{aligned} \end{aligned}$$

(5.34)

Since $\lim_{\tau\rightarrow\infty}e^{-\hat{\sigma}\tau}\tilde{R}^{2}_{t_{0}-\tau}=0$, for any $\varepsilon>0$, we can find $\tau_{0}=\tau_{0}(\varepsilon,\tilde{D},t_{0})\geq 0$ such that

$$\frac{(2+5\varepsilon')}{C^{1}\lambda^{2}\tau}e^{\tilde{\delta}(-\tau)}F(t_{0}-\tau) +\frac{1}{C^{1}\tau}e^{-\tilde{\delta}\tau} \bigl(w_{t}(t_{0}-\tau),w(t_{0}-\tau)\bigr)\leq \varepsilon. $$

Thus we have

$$E_{w}(t_{0})\leq\varepsilon+\phi_{t_{0},\tau}\bigl( \bigl(u_{0}^{1},v_{0}^{1}, \theta_{0}^{1}\bigr),\bigl(u_{0}^{2},v_{0}^{2}, \theta_{0}^{2}\bigr)\bigr)\quad \mbox{for all } \bigl(u_{0}^{i},v_{0}^{i}, \theta_{0}^{i}\bigr)\in\tilde{D}_{t_{0}-\tau_{0}}. $$

By Lemma 3.1 we only need to show that $\phi_{t_{0},\tau_{0}}(\cdot,\cdot)$ defined by (5.34) is a contractive function on $\tilde{D}_{t_{0}-\tau_{0}} \times \tilde{D}_{t_{0}-\tau_{0}}$. Let $(u_{n},u_{nt},\theta_{n})$ be the corresponding solutions of $(u_{0}^{n}, v_{0}^{n}, \theta_{0}^{n})\in\tilde{D}_{t_{0}-\tau_{0}}, n=1,2,\dots$. Since $\tilde{D}_{t_{0}-\tau_{0}}$ is a bounded subset in $X_{0}$, by (4.16) we know that

$$\begin{aligned} \bigl\Vert \bigl(u_{n}(s),u_{nt}(s), \theta_{n}(s)\bigr) \bigr\Vert _{X_{0}}\leq C'_{t_{0},\tau_{0}}< +\infty\quad \mbox{for all } s\in[t_{0}- \tau_{0},t_{0}] \mbox{ and } n\in N, \end{aligned}$$

(5.35)

where $C'_{t_{0},\tau_{0}}$ depends on $t_{0},\tau_{0}$.

Now, we will deal with the right terms in (5.34) one by one.

Without loss of generality, assuming first that

$$u_{n}\rightarrow u\quad \mbox{weak-star in } L^{\infty} \bigl(t_{0}-\tau_{0}, t_{0}; H_{0}^{2}( \Omega)\bigr) $$

and considering that compact embeddings $H_{0}^{2}(\Omega)\hookrightarrow\hookrightarrow H_{0}^{1}(\Omega)$, we have

$$u_{n}\rightarrow u\quad \mbox{strongly in } L^{2} \bigl(t_{0}-\tau_{0}, t_{0}; H_{0}^{1}( \Omega)\bigr), $$

so we obtain

$$\begin{aligned} \lim_{n\rightarrow\infty}\lim_{m\rightarrow\infty} \int_{t_{0}-\tau_{0}}^{t_{0}} e^{\tilde{\delta}\xi} \bigl\Vert \nabla u_{n}(\xi)-\nabla u_{m}(\xi) \bigr\Vert ^{2}\,d \xi =0. \end{aligned}$$

(5.36)

Second, similarly assuming that

$$u_{n}\rightarrow u\quad \mbox{weak-star in } L^{\infty} \bigl(t_{0}-\tau_{0}, t_{0}; H_{0}^{1}( \Omega)\bigr) $$

and considering compact embeddings $H_{0}^{1}(\Omega)\hookrightarrow \hookrightarrow L^{2}(\Omega)$, we also get

$$\begin{aligned} \begin{aligned} &\lim_{n\rightarrow\infty}\lim _{m\rightarrow\infty} \int_{t_{0}-\tau_{0}}^{t_{0}}e^{\tilde{\delta}\xi} \bigl\Vert u_{n}(\xi)- u_{m}(\xi) \bigr\Vert ^{2}\,d\xi =0, \\ &\lim_{n\rightarrow\infty}\lim_{m\rightarrow\infty}\biggl( \int_{t_{0}-\tau_{0}}^{t_{0}}e^{\tilde{\delta}\xi} \bigl\Vert u_{n}(\xi)- u_{m}(\xi) \bigr\Vert ^{2}\,d\xi \biggr)^{\frac{1}{2}}=0. \end{aligned} \end{aligned}$$

(5.37)

Finally, with

$$\begin{aligned} &u_{n}\rightarrow u\quad \mbox{weak-star in } L^{\infty} \bigl(t_{0}-\tau_{0}, t_{0}; L^{2}(\Omega) \bigr), \\ &u_{nt}\rightarrow u_{t}\quad \mbox{weak-star in } L^{\infty}\bigl(t_{0}-\tau_{0}, t_{0}; L^{2}(\Omega)\bigr), \end{aligned}$$

we have

$$u_{n}\rightarrow u\quad \mbox{strongly in } C\bigl(t_{0}- \tau_{0}, t_{0}; L^{2}(\Omega)\bigr). $$

So, it is easy to obtain that

$$\begin{aligned} \begin{aligned}[b] &\lim_{n\rightarrow\infty}\lim _{m\rightarrow\infty} \int_{\Omega} \bigl(u_{nt}(t_{0})-u_{mt}(t_{0}) \bigr) \bigl(u_{n}(t_{0})-u_{m}(t_{0}) \bigr)\,dx \\ &\quad =\lim_{n\rightarrow\infty}\lim_{m\rightarrow\infty} \int_{\Omega} \bigl(u_{nt}(t_{0})-u_{mt}(t_{0}) \bigr) \bigl(u_{n}(t_{0})-u(t_{0})+u(t_{0})-u_{m}(t_{0}) \bigr)\,dx \\ &\quad =\lim_{n\rightarrow\infty}\lim_{m\rightarrow\infty} \int_{\Omega} \bigl(u_{nt}(t_{0})-u_{mt}(t_{0}) \bigr) \bigl(u_{n}(t_{0})-u(t_{0})\bigr)\,dx \\ &\qquad {}+\lim_{n\rightarrow\infty}\lim_{m\rightarrow\infty} \int_{\Omega} \bigl(u_{nt}(t_{0})-u_{mt}(t_{0}) \bigr) \bigl(u(t_{0})-u_{m}(t_{0})\bigr)\,dx \\ &\quad \leq \bigl\Vert u_{nt}(t_{0})-u_{mt}(t_{0}) \bigr\Vert _{L^{\infty}(L^{2}(\Omega))} \bigl\Vert u_{n}(t_{0})-u(t_{0}) \bigr\Vert \\ &\qquad {}+ \bigl\Vert u_{nt}(t_{0})-u_{mt}(t_{0}) \bigr\Vert _{L^{\infty}(L^{2}(\Omega))} \bigl\Vert u(t_{0})-u_{m}(t_{0}) \bigr\Vert \\ &\quad \leq C\bigl[ \bigl\Vert u_{n}(t_{0})-u(t_{0}) \bigr\Vert + \bigl\Vert u(t_{0})-u_{m}(t_{0}) \bigr\Vert \bigr] \\ &\quad \rightarrow 0. \end{aligned} \end{aligned}$$

(5.38)

By assumptions (1.6) on $g(\cdot)$, using the embedding theorem combined with (5.35), we have

$$\begin{aligned} \begin{aligned}[b] &\lim_{n\rightarrow\infty}\lim _{m\rightarrow\infty} \int_{t_{0}-\tau_{0}}^{t_{0}} e^{\tilde{\delta}\xi} \int_{\Omega}\bigl(u_{nt}(\xi)-u_{mt}(\xi) \bigr) \bigl(g\bigl(u_{n}(\xi)\bigr)-g\bigl(u_{m}(\xi)\bigr) \bigr)\,dx\,d\xi \\ &\quad \leq \lim_{n\rightarrow\infty}\lim_{m\rightarrow\infty} \int_{t_{0}-\tau_{0}}^{t_{0}} e^{\tilde{\delta}\xi} \int_{\Omega}\bigl(u_{nt}(\xi)-u_{mt}(\xi) \bigr)k_{2}\bigl(u_{n}(\xi)-u_{m}(\xi)\bigr)\\ &\qquad {}\times \bigl(1+ \bigl\vert u_{n}(\xi) \bigr\vert ^{\rho-1}+ \bigl\vert u_{m}(\xi) \bigr\vert ^{\rho-1}\bigr)\,dx\,d\xi \\ &\quad \leq C'_{t_{0},\tau_{0}} \lim_{n\rightarrow\infty}\lim _{m\rightarrow\infty}\biggl( \int_{t_{0}-\tau_{0}}^{t_{0}} e^{\tilde{\delta}\xi} \bigl\Vert \bigl(u_{n}(\xi)-u_{m}(\xi)\bigr) \bigr\Vert ^{2}\,d\xi \biggr)^{\frac{1}{2}} \\ &\quad =0. \end{aligned} \end{aligned}$$

(5.39)

Similarly, since $\int_{s}^{t_{0}}e^{\tilde{\delta}\xi}\int_{\Omega} (u_{nt}(\xi)-u_{mt}(\xi))(g(u_{n}(\xi))-g(u_{m}(\xi)))\,dx\,d\xi$ is bounded for each $s\in[\tau,t_{0}]$, by (5.39) and the Lebesgue dominated convergence theorem we have

$$\begin{aligned} &\lim_{n\rightarrow\infty}\lim _{m\rightarrow\infty} \int_{t_{0}-\tau_{0}}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi} \int_{\Omega} \bigl(u_{nt}(\xi)-u_{mt}(\xi) \bigr) \bigl(g\bigl(u_{n}(\xi)\bigr)-g\bigl(u_{m}(\xi)\bigr) \bigr)\,dx\,d\xi \,ds \\ &\quad = \int_{t_{0}-\tau_{0}}^{t_{0}}\biggl(\lim_{n\rightarrow\infty}\lim _{m\rightarrow\infty} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi} \int_{\Omega} \bigl(u_{nt}(\xi)-u_{mt}(\xi) \bigr) \bigl(g\bigl(u_{n}(\xi)\bigr)-g\bigl(u_{m}(\xi)\bigr) \bigr)\,dx\,d\xi \biggr)\,ds \\ &\quad = \int_{t_{0}-\tau_{0}}^{t_{0}}0\,ds \\ &\quad =0. \end{aligned}$$

(5.40)

Combining (5.36)-(5.40), we get that $\Phi_{t_{0},\tau_{0}}(\cdot,\cdot)$ is a contractive function on $\tilde{D}_{t_{0}-\tau_{0}}\times\tilde{D}_{t_{0}-\tau_{0}}$. The proof is finished by Lemma 3.1. □