Pullback attractor for N-dimensional thermoelastic coupled structure equations

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

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 \(N=1,2\).

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

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

for every \(t>0\), and the initial conditions

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

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 \(t\in R\),

and

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).

We first introduce the following abbreviations:

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

equipped with the norm

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.

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

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

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

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

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

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

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

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

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

and

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

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

where \(C_{0}\geq1\). So

By Young’s inequality we have

and

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

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

By Young’s inequality we have

and

So

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

and

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

From (4.6) we have

Note that

so by (4.9) we have

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

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

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

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

So from (4.13)-(4.14) we get

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

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

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

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

If we set

and

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

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

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

and

First, we have

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

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

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

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

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

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

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

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

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

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

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

By the value theorem and (4.19) we have

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

Substituting (5.29) into (5.25), we obtain

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

So from (5.30) we obtain

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

By (5.3) we have

We set

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

Thus we have

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

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

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

so we obtain

Second, similarly assuming that

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

Finally, with

we have

So, it is easy to obtain that

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

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

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. □

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).

Affiliations

1. Department of Mathematics, Taiyuan University of Technology, Taiyuan, 030024, P.R. China

   • Danxia Wang

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

   • Yinzhu Wang

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.

Corresponding author

Correspondence to Danxia Wang.

Competing interests

The authors declare that they have no competing interests.

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