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Boundary Value Problems
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Pullback attractor for N-dimensional thermoelastic coupled structure equations

Boundary Value Problems20182018:5

https://doi.org/10.1186/s13661-017-0921-7

©  The Author(s) 2018

  • Received: 17 October 2017
  • Accepted: 7 December 2017
  • Published:

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 \(N=1,2\).

Keywords

  • pullback attractor
  • pullback asymptotically compact
  • contractive function
  • thermoelastic coupled structure equations

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. (1)

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

     
  2. (2)

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

     
  3. (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. (1)

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

     
  2. (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. (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. (1)

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

     
  2. (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)
  1. (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 \(0<\varepsilon'<\min\{\frac{\sqrt{(9+\lambda_{0}^{2})^{2}+40\eta\lambda_{0}^{2}}-(9+\lambda_{0}^{2})}{20},\eta\}\).
     
  2. (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)
     
  3. (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)
     
  4. (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. □

Declarations

Acknowledgements

The project is supported by the National Youth Fund of China (Grants Nos. 11401422 and 11401420), the Natural Science Fund of Shanxi Province, China (Grants Nos. 2015011006 and 201701D121010).

Authors’ contributions

This paper was mainly completed by DX. YZ gave the exact conditions of the source term \(g(u)\) and dealed with it through the paper. YZ also dealed with the right terms in (5.34). All authors read and approved the final manuscript.

Competing interests

The authors declare that they have no competing interests.

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
Department of Mathematics, Taiyuan University of Technology, Taiyuan, 030024, P.R. China
(2)
Department of Mathematics, Taiyuan University of Science and Technology, Taiyuan, 030024, P.R. China

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