In this section, we prove the pullback attractor in \(X_{0}\).
Theorem 5.1
Assume that assumptions (H1)-(H3) of
\(g(\cdot)\)
hold and that
\(h(x,t), q(x,t)\in L^{2}_{\mathrm{loc}}(R,H)\)
satisfy (1.10) and (1.11) with some
δ̃
satisfying
\(0<\tilde{\delta}<\min\{\frac{\eta}{2},2\lambda_{0}^{2},\frac{\eta-\varepsilon'}{2+5\varepsilon'},\delta\}\) (\(0<\varepsilon'<\min\{\frac{\sqrt{(9+\lambda_{0}^{2})^{2}+40\eta\lambda_{0}^{2}}-(9+\lambda_{0}^{2})}{20},\eta\}\)). Then there exists a pullback attractor
\(A=\{A_{t}\}_{t\in R}\)
in
\(X_{0}\)
for the nonautonomous dynamical system
\((\theta,\Phi)\)
defined by (4.1).
Proof
Fix \(t_{0}\in R\). Let \(y_{i}(t)=(u_{i}(t),v_{i}(t),\theta_{i}(t))\)
\((i=1,2)\) be the corresponding weak solution to \(y_{0}^{i}=(u_{0}^{i},v_{0}^{i},\theta_{0}^{i})\in \tilde{D}_{t_{0}-\tau}\), where \(\tau\geq0\), and let \(w(t)=u_{1}(t)-u_{2}(t)\), \(\tilde{\theta}(t)=\theta_{1}(t)-\theta_{2}(t)\). Then \((w,\tilde{\theta})\) satisfy
$$\begin{aligned} & w_{tt}+\alpha\triangle^{2}w-\beta\triangle w -\biggl(\sigma\biggl( \int_{\Omega}(\nabla u_{1})^{2}\,dx\biggr) \triangle u_{1} -\sigma\biggl( \int_{\Omega}(\nabla u_{2})^{2}\,dx\biggr) \triangle u_{2}\biggr) \\ &\qquad {} +\gamma\triangle \tilde{\theta} +\triangle g + \eta w_{t} =0, \end{aligned}$$
(5.1)
$$\begin{aligned} & \tilde{\theta}_{t}-\triangle \tilde{\theta}-\gamma\triangle w_{t}=0 \end{aligned}$$
(5.2)
with the initial condition \((w(0),w_{t}(0),\tilde{\theta}(0))=(u_{0}^{1},v_{0}^{1},\theta_{0}^{1})-(u_{0}^{2},v_{0}^{2},\theta_{0}^{2})\), where \(\triangle g=g(u_{1})-g(u_{2})\).
Define
$$E_{u}(t)=\frac{1}{2}\bigl( \Vert u_{t} \Vert ^{2}+ \Vert \triangle u \Vert ^{2}+ \Vert \theta \Vert ^{2}\bigr)=\frac{1}{2} \bigl\Vert \phi(t-t_{0}+ \tau, t_{0}-\tau,y_{0}) \bigr\Vert ^{2}_{X_{0}} $$
and
$$F(t)=\frac{1}{2}\bigl( \Vert w_{t} \Vert ^{2}+ \alpha \Vert \triangle w \Vert ^{2}+\beta \Vert \nabla w \Vert ^{2}+\sigma\bigl( \Vert \nabla u_{1} \Vert ^{2} \bigr) \Vert \nabla w \Vert ^{2}+ \Vert \tilde{\theta} \Vert ^{2}\bigr). $$
First, we have
$$\begin{aligned} F(t)&\geq\frac{1}{2}\bigl( \Vert w_{t} \Vert ^{2}+\alpha \Vert \triangle w \Vert ^{2}+ \Vert \tilde{ \theta} \Vert ^{2}\bigr) \\ &\geq\frac{1}{2}C^{1}\bigl( \Vert w_{t} \Vert ^{2}+ \Vert \triangle w \Vert ^{2}+ \Vert \tilde{ \theta} \Vert ^{2}\bigr) \\ &= C^{1}E_{w}(t), \end{aligned}$$
(5.3)
where \(C^{1}=\min\{1,\alpha\}\).
Multiplying (5.1) by \(e^{\tilde{\delta}t}w_{t}\) and (5.2) by \(e^{\tilde{\delta}t}\tilde{\theta}\) and then summing, we obtain
$$\begin{aligned} & \frac{d}{dt}\bigl[e^{\tilde{\delta}t}F(t)\bigr] + e^{\tilde{\delta} t}\bigl(\eta \Vert w_{t} \Vert ^{2}+ \Vert \nabla \tilde{\theta} \Vert ^{2}\bigr) \\ &\quad =\tilde{\delta}e^{\tilde{\delta}t}F(t) \\ &\qquad {}-e^{\tilde{\delta}t}\biggl( \int_{\Omega}\triangle g w_{t}\,dx - \sigma' \bigl( \Vert \nabla u_{1} \Vert ^{2}\bigr) \int_{\Omega}\nabla u_{1}\nabla u_{1t}\,dx \Vert \nabla w \Vert ^{2} - \int_{\Omega}\triangle\sigma w_{t}\,dx\biggr), \end{aligned}$$
(5.4)
where \(\triangle\sigma=\sigma( \Vert \nabla u_{1} \Vert ^{2})-\sigma( \Vert \nabla u_{2} \Vert ^{2})\), and δ̃ satisfies (4.18).
Integrating (5.4) from s to \(t_{0}\), we have
$$\begin{aligned} &e^{\tilde{\delta}t_{0}}F(t_{0})-e^{\tilde{\delta}s}F(s)+ \int_{s}^{t_{0}}e^{\tilde{\delta}\xi}\bigl(\eta \Vert w_{t} \Vert ^{2}+ \Vert \nabla \tilde{\theta} \Vert ^{2}\bigr)\,d\xi \\ &\quad =\tilde{\delta} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi}F(\xi)\,d\xi - \int_{s}^{t_{0}}e^{\tilde{\delta}\xi}\biggl( \int_{\Omega}\triangle g w_{t}\,dx\biggr)\,d\xi \\ &\qquad {}+ \int_{s}^{t_{0}}e^{\tilde{\delta}\xi}\biggl( \sigma'\bigl( \Vert \nabla u_{1} \Vert ^{2} \bigr) \int_{\Omega}\nabla u_{1}\nabla u_{1t}\,dx \Vert \nabla w \Vert ^{2} + \int_{\Omega}\triangle\sigma w_{t}\,dx\biggr)\,d\xi . \end{aligned}$$
(5.5)
Integrating (5.5) from \(t_{0}-\tau\) to \(t_{0}\) with respect to s, we obtain
$$\begin{aligned} &\tau e^{\tilde{\delta}t_{0}}F(t_{0})- \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}s}F(s)\,ds + \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi}\bigl(\eta \Vert w_{t} \Vert ^{2}+ \Vert \nabla \tilde{\theta} \Vert ^{2}\bigr)\,d\xi \,ds \\ &\quad =\tilde{\delta} \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi}F(\xi)\,d\xi \,ds - \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi}\biggl( \int_{\Omega}\triangle g w_{t}\,dx\biggr)\,d\xi \,ds \\ &\qquad {}+ \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi}\biggl( \sigma'\bigl( \Vert \nabla u_{1} \Vert ^{2} \bigr) \int_{\Omega}\nabla u_{1}\nabla u_{1t}\,dx \Vert \nabla w \Vert ^{2} + \int_{\Omega}\triangle\sigma w_{t}\,dx\biggr)\,d\xi \,ds. \end{aligned}$$
(5.6)
Similarly, multiplying (5.1) by \(e^{\tilde{\delta}t}w\), we have
$$\begin{aligned} &\frac{d}{dt}\bigl[e^{\tilde{\delta}t}(w_{t},w) \bigr]+e^{\tilde{\delta}t}\bigl(\alpha \Vert \triangle w \Vert ^{2}+ \beta \Vert \nabla w \Vert ^{2}+\sigma\bigl( \Vert \nabla u_{1} \Vert ^{2}\bigr) \Vert \nabla w \Vert ^{2}\bigr) \\ &\quad =(\tilde{\delta}-\eta)e^{\tilde{\delta}t}(w_{t},w)+e^{\tilde{\delta}t} \Vert w_{t} \Vert ^{2} \\ &\qquad {}-e^{\tilde{\delta}t}\biggl( \int_{\Omega}\triangle g w\,dx -\gamma \int_{\Omega}\triangle \tilde{\theta} w\,dx - \int_{\Omega}\triangle\sigma \triangle u_{2} w \,dx \biggr). \end{aligned}$$
(5.7)
First, integrating (5.7) over \([s,t_{0}]\), we get that
$$\begin{aligned} \begin{aligned}[b] & \int_{s}^{t_{0}}e^{\tilde{\delta}\xi}\bigl(\alpha \Vert \triangle w \Vert ^{2}+\beta \Vert \nabla w \Vert ^{2}+ \sigma\bigl( \Vert \nabla u_{1} \Vert ^{2}\bigr) \Vert \nabla w \Vert ^{2}\bigr)\,d\xi + e^{\tilde{\delta}t_{0}} \bigl(w_{t}(t_{0}),w(t_{0})\bigr) \\ &\quad =e^{\tilde{\delta}s}\bigl(w_{t}(s),w(s)\bigr)+(\tilde{\delta}-\eta) \int_{s}^{t_{0}}e^{\tilde{\delta}\xi}(w_{t},w)\,d \xi + \int_{s}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w_{t} \Vert ^{2}\,d\xi \\ &\qquad {}- \int_{s}^{t_{0}} \int_{\Omega}e^{\tilde{\delta}\xi}\triangle g w\,dx\,d\xi -\gamma \int_{s}^{t_{0}} \int_{\Omega}e^{\tilde{\delta}\xi}\triangle \tilde{\theta} w\,dx\,d\xi\\ &\qquad {} - \int_{s}^{t_{0}} \int_{\Omega}e^{\tilde{\delta}\xi}\triangle\sigma \triangle u_{2} w \,dx\,d\xi . \end{aligned} \end{aligned}$$
(5.8)
Then integrating (5.8) over \([t_{0}-\tau,t_{0}]\) with respect to s, we obtain
$$\begin{aligned} &\tilde{\delta} \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi}\bigl(\alpha \Vert \triangle w \Vert ^{2}+\beta \Vert \nabla w \Vert ^{2}+ \sigma\bigl( \Vert \nabla u_{1} \Vert ^{2}\bigr) \Vert \nabla w \Vert ^{2}\bigr)\,d\xi \,ds \\ &\quad =-\tilde{\delta}\tau e^{\tilde{\delta}t_{0}}\bigl(w_{t}(t_{0}),w(t_{0}) \bigr) + \tilde{\delta} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}s}\bigl(w_{t}(s),w(s) \bigr)\,ds \\ &\qquad {}+\tilde{\delta}(\tilde{\delta}-\eta) \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi}(w_{t},w)\,d \xi \,ds +\tilde{\delta} \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w_{t} \Vert ^{2}\,d\xi \,ds \\ &\qquad {}-\tilde{\delta} \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}} \int_{\Omega}e^{\tilde{\delta}\xi}\triangle g w\,dx\,d\xi \,ds -\gamma \tilde{\delta} \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}} \int_{\Omega}e^{\tilde{\delta}\xi}\triangle \tilde{\theta} w\,dx\,d\xi \,ds \\ &\qquad {}-\tilde{\delta} \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}} \int_{\Omega}e^{\tilde{\delta}\xi}\triangle\sigma \triangle u_{2} w \,dx\,d\xi \,ds. \end{aligned}$$
(5.9)
Substituting (5.9) into (5.6), we deduce that
$$\begin{aligned} &\tau e^{\tilde{\delta}t_{0}}F(t_{0})- \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}s}F(s)\,ds + \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi}\bigl(\eta \Vert w_{t} \Vert ^{2}+ \Vert \nabla \tilde{\theta} \Vert ^{2}\bigr)\,d\xi \,ds \\ &\quad =\tilde{\delta} \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi} \frac{1}{2} \bigl( \Vert w_{t} \Vert ^{2}+\alpha \Vert \triangle w \Vert ^{2}+\beta \Vert \nabla w \Vert ^{2}+\sigma\bigl( \Vert \nabla u_{1} \Vert ^{2}\bigr) \Vert \nabla w \Vert ^{2}+ \Vert \tilde{\theta} \Vert ^{2}\bigr) \,d\xi \,ds \\ &\qquad {}- \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi}\biggl( \int_{\Omega}\triangle g w_{t}\,dx - \sigma' \bigl( \Vert \nabla u_{1} \Vert ^{2}\bigr) \int_{\Omega}\nabla u_{1}\nabla u_{1t}\,dx \Vert \nabla w \Vert ^{2} \\ &\qquad {}- \int_{\Omega}\triangle\sigma w_{t}\,dx\biggr)\,d\xi \,ds \\ &\quad =\tilde{\delta} \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi} \frac{1}{2} \bigl( \Vert w_{t} \Vert ^{2}+ \Vert \tilde{\theta} \Vert ^{2}\bigr)\,d\xi \,ds -\frac{1}{2}\tilde{\delta}\tau e^{\tilde{\delta}t_{0}}\bigl(w_{t}(t_{0}),w(t_{0})\bigr) \\ &\qquad {}+ \frac{1}{2}\tilde{\delta} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}s}\bigl(w_{t}(s),w(s) \bigr)\,ds +\frac{1}{2}\tilde{\delta}(\tilde{\delta}-\eta) \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi}(w_{t},w)\,d \xi \,ds \\ &\qquad {}+\frac{1}{2}\tilde{\delta} \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w_{t} \Vert ^{2}\,d\xi \,ds -\frac{1}{2}\tilde{\delta} \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}} \int_{\Omega}e^{\tilde{\delta}\xi}\triangle g w\,dx\,d\xi \,ds \\ &\qquad {}-\frac{1}{2}\gamma\tilde{\delta} \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}} \int_{\Omega}e^{\tilde{\delta}\xi}\triangle \tilde{\theta} w\,dx\,d\xi \,ds -\frac{1}{2}\tilde{\delta} \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}} \int_{\Omega}e^{\tilde{\delta}\xi}\triangle\sigma \triangle u_{2} w \,dx\,d\xi \,ds \\ &\qquad {}- \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi}\biggl( \int_{\Omega}\triangle g w_{t}\,dx - \sigma' \bigl( \Vert \nabla u_{1} \Vert ^{2}\bigr) \int_{\Omega}\nabla u_{1}\nabla u_{1t}\,dx \Vert \nabla w \Vert ^{2} \\ &\qquad {}- \int_{\Omega}\triangle\sigma w_{t}\,dx\biggr)\,d\xi \,ds. \end{aligned}$$
(5.10)
Second, integrating (5.7) from \(t_{0}-\tau\) to \(t_{0}\), we have
$$\begin{aligned} & \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi}\bigl( \Vert w_{t} \Vert ^{2}+\alpha \Vert \triangle w \Vert ^{2}+\beta \Vert \nabla w \Vert ^{2}+\sigma\bigl( \Vert \nabla u_{1} \Vert ^{2}\bigr) \Vert \nabla w \Vert ^{2}+ \Vert \tilde{\theta} \Vert ^{2}\bigr)\,d\xi \\ &\quad =e^{\tilde{\delta}(t_{0}-\tau)}\bigl(w_{t}(t_{0}-\tau),w(t_{0}- \tau)\bigr)-e^{\tilde{\delta}t_{0}}\bigl(w_{t}(t_{0}),w(t_{0}) \bigr) \\ &\qquad {}+(\tilde{\delta}-\eta) \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi}(w_{t},w)\,d \xi + \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w_{t} \Vert ^{2}\,d\xi \\ &\qquad {}- \int_{t_{0}-\tau}^{t_{0}} \int_{\Omega}e^{\tilde{\delta}\xi}\triangle g w\,dx \,d\xi -\gamma \int_{t_{0}-\tau}^{t_{0}} \int_{\Omega}e^{\tilde{\delta}\xi}\triangle \tilde{\theta} w\,dx\,d\xi \\ &\qquad {} - \int_{t_{0}-\tau}^{t_{0}} \int_{\Omega}e^{\tilde{\delta}\xi}\triangle\sigma \triangle u_{2} w \,dx\,d\xi + \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi}\bigl( \Vert w_{t} \Vert ^{2}+ \Vert \tilde{\theta} \Vert ^{2}\bigr)\,d\xi . \end{aligned}$$
(5.11)
Substituting (5.11) into (5.10) and noting that \(\tilde{\delta}<\frac{\eta}{2}\), we have
$$\begin{aligned} &\tau e^{\tilde{\delta}t_{0}}F(t_{0})+ \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}s}F(s)\,ds + \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi} \Vert \nabla \tilde{ \theta} \Vert ^{2}\,d\xi \,ds \\ &\quad \leq\frac{\tilde{\delta}}{2} \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi} \Vert \tilde{\theta} \Vert ^{2}\,d\xi \,ds -\biggl(\frac{1}{2}\tilde{\delta}\tau+1 \biggr) e^{\tilde{\delta}t_{0}}\bigl(w_{t}(t_{0}),w(t_{0}) \bigr) \\ &\qquad {}+ \frac{1}{2}\tilde{\delta} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}s}\bigl(w_{t}(s),w(s) \bigr)\,ds +\frac{1}{2}\tilde{\delta}(\tilde{\delta}-\eta) \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi}(w_{t},w)\,d \xi \,ds \\ &\qquad {}-\frac{1}{2}\tilde{\delta} \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}} \int_{\Omega}e^{\tilde{\delta}\xi}\triangle g w\,dx\,d\xi \,ds - \frac{1}{2}\gamma\tilde{\delta} \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}} \int_{\Omega}e^{\tilde{\delta}\xi}\triangle \tilde{\theta} w\,dx\,d\xi \,ds \\ &\qquad {}-\frac{1}{2}\tilde{\delta} \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}} \int_{\Omega}e^{\tilde{\delta}\xi}\triangle\sigma\triangle u_{2} w \,dx\,d\xi \,ds + \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi}\biggl( \int_{\Omega}\triangle g w_{t}\,dx\biggr)\,d\xi \,ds \\ &\qquad {}- \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi}\biggl( \sigma'\bigl( \Vert \nabla u_{1} \Vert ^{2} \bigr) \int_{\Omega}\nabla u_{1}\nabla u_{1t}\,dx \Vert \nabla w \Vert ^{2} + \int_{\Omega}\triangle\sigma w_{t}\,dx\biggr)\,d\xi \,ds \\ &\qquad {}+e^{\tilde{\delta}(t_{0}-\tau)}\bigl(w_{t}(t_{0}-\tau),w(t_{0}- \tau)\bigr) +(\tilde{\delta}-\eta) \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi}(w_{t},w)\,d \xi + \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w_{t} \Vert ^{2}\,d\xi \\ &\qquad {}- \int_{t_{0}-\tau}^{t_{0}} \int_{\Omega}e^{\tilde{\delta}\xi}\triangle g w\,dx \,d\xi -\gamma \int_{t_{0}-\tau}^{t_{0}} \int_{\Omega}e^{\tilde{\delta}\xi}\triangle \tilde{\theta} w\,dx\,d\xi \\ &\qquad {} - \int_{t_{0}-\tau}^{t_{0}} \int_{\Omega}e^{\tilde{\delta}\xi}\triangle\sigma \triangle u_{2} w \,dx\,d\xi + \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi}\bigl( \Vert w_{t} \Vert ^{2}+ \Vert \tilde{\theta} \Vert ^{2}\bigr)\,d\xi . \end{aligned}$$
(5.12)
-
(I)
Using the Schwarz and Young inequalities, for \(t_{0}-\tau\leq s< t_{0}\), we have
$$\begin{aligned} &\frac{1}{2}\tilde{\delta} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}s}\bigl(w_{t}(s),w(s) \bigr)\,ds \leq\frac{\tilde{\delta}^{2}}{8\varepsilon'} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}s} \bigl\Vert w(s) \bigr\Vert ^{2}\,ds +\varepsilon' \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}s} \bigl\Vert w_{t}(s) \bigr\Vert ^{2}\,ds, \end{aligned}$$
(5.13)
$$\begin{aligned} &\begin{aligned}[b] &\frac{1}{2}\tilde{\delta}(\tilde{ \delta}-\eta) \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi}(w_{t},w)\,d \xi \,ds \\ &\quad \leq \frac{1}{2}\tau\tilde{\delta}(\tilde{\delta}-\eta) \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w_{t} \Vert \Vert w \Vert \,d\xi \\ &\quad \leq \frac{[\frac{1}{2}\tau\tilde{\delta}(\tilde{\delta}-\eta)]^{2}}{4\varepsilon'} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w \Vert ^{2}\,d\xi +\varepsilon' \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w_{t} \Vert ^{2}\,d\xi , \end{aligned} \end{aligned}$$
(5.14)
$$\begin{aligned} & \begin{aligned}[b] & {-}\frac{1}{2}\gamma\tilde{\delta} \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}} \int_{\Omega}e^{\tilde{\delta}\xi}\triangle \tilde{\theta} w\,dx\,d\xi \,ds\\ &\quad \leq \frac{\varepsilon'}{\lambda_{0}^{2}} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert \nabla\tilde{ \theta} \Vert ^{2}\,d\xi +\frac{(\tau \frac{1}{2}\gamma\tilde{\delta})^{2}}{4\varepsilon'} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert \nabla w \Vert ^{2}\,d\xi , \end{aligned} \end{aligned}$$
(5.15)
$$\begin{aligned} &(\tilde{\delta}-\eta) \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi}(w_{t},w)\,d \xi \leq\frac{(\tilde{\delta}-\eta)^{2}}{8\varepsilon'} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\sigma}\xi} \Vert w \Vert ^{2}\,d\xi +2\varepsilon' \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\sigma}\xi} \Vert w_{t} \Vert ^{2}\,d\xi , \end{aligned}$$
(5.16)
and
$$\begin{aligned} -\gamma \int_{t_{0}-\tau}^{t_{0}} \int_{\Omega}e^{\tilde{\delta}\xi}\triangle \tilde{\theta} w\,dx\,d\xi \leq \frac{\varepsilon'}{\lambda_{0}^{2}} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert \nabla \tilde{ \theta} \Vert ^{2} \,d\xi +\frac{\lambda_{0}^{2}\gamma^{2}}{4\varepsilon'} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert \nabla w \Vert ^{2}\,d\xi , \end{aligned}$$
(5.17)
where \(0<\varepsilon'<\min\{\frac{\sqrt{(9+\lambda_{0}^{2})^{2}+40\eta\lambda_{0}^{2}}-(9+\lambda_{0}^{2})}{20},\eta\}\).
-
(II)
By assumption (1.6) on \(g(\cdot)\), the Hölder inequality, and the embedding theorem combined with (4.19), for \(t_{0}-\tau\leq s< t_{0}\), we have
$$\begin{aligned} &\frac{1}{2}\tilde{\delta} \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}} \int_{\Omega}e^{\tilde{\delta}\xi}\triangle g w\,dx\,d\xi \,ds \\ &\quad \leq\frac{1}{2}\tilde{\delta}\tau\biggl( \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \int_{\Omega} \bigl\vert g(u_{2})-g(u_{1}) \bigr\vert ^{2}\,dx\,d\xi \biggr)^{\frac{1}{2}}\biggl( \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w \Vert ^{2}\,d\xi \biggr)^{\frac{1}{2}} \\ &\quad \leq \frac{1}{2}\tilde{\delta}C\tau\biggl( \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \int_{\Omega}\bigl(1+ \vert u_{1} \vert ^{2\rho-2}+ \vert u_{2} \vert ^{2\rho-2}\bigr) \vert w \vert ^{2}\,dx\,d\xi \biggr)^{\frac{1}{2}} \\ &\qquad {}\times\biggl( \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w \Vert ^{2}\,d\xi \biggr)^{\frac{1}{2}} \\ &\quad \leq \frac{1}{2}\tilde{\delta}C\tau\biggl( \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \int_{\Omega}\bigl( \vert u_{1} \vert ^{2}+ \vert u_{2} \vert ^{2}+ \vert u_{1} \vert ^{2\rho} + \vert u_{2} \vert ^{2\rho}\bigr)\,dx\,d\xi \biggr)^{\frac{1}{2}} \\ &\qquad {}\times\biggl( \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w \Vert ^{2}\,d\xi \biggr)^{\frac{1}{2}} \\ &\quad \leq \frac{1}{2}\tilde{\delta}C\tau\biggl( \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi}\bigl( \Vert \nabla u_{1} \Vert ^{2}+ \Vert \nabla u_{2} \Vert ^{2} + \Vert \nabla u_{1} \Vert ^{2\rho}+ \Vert \nabla u_{2} \Vert ^{2\rho}\bigr)\,d\xi \biggr)^{\frac{1}{2}} \\ &\qquad {}\times \biggl( \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w \Vert ^{2}\,d\xi \biggr)^{\frac{1}{2}} \\ &\quad \leq C_{t_{0},\tau}\biggl( \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w \Vert ^{2}\,d\xi \biggr)^{\frac{1}{2}}, \end{aligned}$$
(5.18)
and similarly we also have
$$\begin{aligned} - \int_{t_{0}-\tau}^{t_{0}} \int_{\Omega}e^{\tilde{\delta}\xi}\triangle g w\,dx \,d\xi \leq C_{t_{0},\tau}\biggl( \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\sigma}\xi} \Vert w \Vert ^{2}\,d\xi \biggr)^{\frac{1}{2}}. \end{aligned}$$
(5.19)
-
(III)
By the value theorem, (4.18), the continuity of \(\sigma(\cdot)\), and the Schwarz inequality, for \(t_{0}-\tau\leq s< t_{0}\), we obtain that
$$\begin{aligned} &-\frac{1}{2}\tilde{\delta} \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}} \int_{\Omega}e^{\tilde{\delta}\xi}\triangle\sigma\triangle u_{2} w \,dx\,d\xi \,ds \\ &\quad \leq\frac{1}{2}\tilde{\delta}\tau \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi}\sigma'( \xi_{1}) \bigl( \Vert \triangle u_{1} \Vert ^{2}- \Vert \triangle u_{2} \Vert ^{2}\bigr) \Vert \triangle u_{2} \Vert \Vert w \Vert \,d\xi \\ &\quad \leq C_{t_{0},\tau} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w \Vert ^{2} \,d\xi , \end{aligned}$$
(5.20)
where \(\xi_{1}\) is between \(\Vert \nabla u_{1} \Vert ^{2}\) and \(\Vert \nabla u_{2} \Vert ^{2}\). Similarly, we have
$$\begin{aligned} & \int_{t_{0}-\tau}^{t_{0}} \int_{\Omega}e^{\tilde{\delta}\xi}\triangle\sigma \triangle u_{2} w \,dx\,d\xi \leq C_{t_{0},\tau} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w \Vert ^{2} \,d\xi , \end{aligned}$$
(5.21)
$$\begin{aligned} &{-} \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi} \sigma' \bigl( \Vert \nabla u_{1} \Vert ^{2}\bigr) \int_{\Omega}\nabla u_{1}\nabla u_{1t}\,dx \Vert \nabla w \Vert ^{2}\,d\xi \,ds \\ &\quad \leq C \tau C_{t_{0},\tau} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert \nabla w \Vert ^{2}\,d\xi , \end{aligned}$$
(5.22)
and
$$\begin{aligned} & {-} \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi} \int_{\Omega}\triangle\sigma w_{t}\,dx\,d\xi \,ds \\ &\quad \leq \tau C_{t_{0},\tau} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w \Vert \Vert w_{t} \Vert \,d\xi \\ &\quad \leq \frac{(\tau C_{t_{0},\tau})^{2}}{4\varepsilon'} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w \Vert ^{2}\,d\xi +\varepsilon' \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w_{t} \Vert ^{2}\,d\xi . \end{aligned}$$
(5.23)
-
(IV)
Since \(\tilde{\delta}\leq2\lambda_{0}^{2}\), we have
$$\begin{aligned} \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi} \Vert \nabla \tilde{ \theta} \Vert ^{2}\,d\xi \,ds &\geq\lambda_{0}^{2} \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi} \Vert \tilde{\theta} \Vert ^{2}\,d\xi \\ &\geq\frac{\tilde{\delta}}{2} \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi} \Vert \tilde{\theta} \Vert ^{2}\,d\xi \,ds. \end{aligned}$$
(5.24)
So, substituting (5.13)-(5.23) into (5.12), by (5.24) we obtain
$$\begin{aligned} &\tau e^{\tilde{\delta}t_{0}}F(t_{0})+ \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}s}F(s)\,ds \\ &\quad \leq \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert \tilde{\theta} \Vert ^{2}\,d\xi -\biggl(\frac{1}{2}\tilde{\delta}\tau+1 \biggr) e^{\tilde{\delta}t_{0}}\bigl(w_{t}(t_{0}),w(t_{0}) \bigr) \\ &\qquad {}+\biggl(\frac{\tilde{\delta}^{2}}{8\varepsilon'}+\frac{[\frac{1}{2}\tau\tilde{\delta} (\tilde{\delta}-\eta)]^{2}}{4\varepsilon'}+\frac{(\tau C_{t_{0},\tau})^{2}}{4\varepsilon'}+ \frac{(\tilde{\delta}-\eta)^{2}}{8\varepsilon'}+2C_{t_{0},\tau}\biggr) \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w \Vert ^{2}\,d\xi \\ &\qquad {}+\bigl(2+5\varepsilon'\bigr) \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w_{t} \Vert ^{2}\,d\xi +2C_{t_{0},\tau}\biggl( \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w \Vert ^{2}\,d\xi \biggr)^{\frac{1}{2}} \\ & \qquad {}+\frac{2\varepsilon'}{\lambda_{0}^{2}} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert \nabla\tilde{ \theta} \Vert ^{2}\,d\xi \\ & \qquad {} +\biggl(\frac{(\tau \frac{1}{2}\gamma\tilde{\delta})^{2}}{4\varepsilon'}+ \frac{\lambda_{0}^{2}\gamma^{2}}{4\varepsilon'}+C \tau C_{t_{0},\tau}+\tau C_{t_{0},\tau}\biggr) \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert \nabla w \Vert ^{2}\,d\xi \\ &\qquad {}+ \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi} \int_{\Omega}\triangle g w_{t}\,dx\,d\xi \,ds+e^{\tilde{\delta}(t_{0}-\tau)}\bigl(w_{t}(t_{0}-\tau),w(t_{0}- \tau)\bigr). \end{aligned}$$
(5.25)
On the other hand, integrating (5.4) over \([t_{0}-\tau,t_{0}]\), we get that
$$\begin{aligned} & e^{\tilde{\delta}t_{0}}F(t_{0})-e^{\tilde{\delta}(t_{0}-\tau)}F(t_{0}- \tau) + \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta} \xi}\bigl(\eta \Vert w_{t} \Vert ^{2}+ \Vert \nabla \tilde{\theta} \Vert ^{2}\bigr)\,d\xi \\ &\quad =\tilde{\delta} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi}F(\xi)\,d\xi - \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \int_{\Omega}\triangle g w_{t}\,dx\,d\xi \\ &\qquad {}- \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \biggl( \sigma'\bigl( \Vert \nabla u \Vert ^{2}\bigr) \int_{\Omega}\nabla u_{1}\nabla u_{1t}\,dx \Vert \nabla w \Vert ^{2} - \int_{\Omega}\triangle\sigma w_{t}\,dx\biggr)\,d\xi . \end{aligned}$$
(5.26)
By the continuity of \(\sigma'(\cdot)\) combined with (4.19) we get
$$\begin{aligned} - \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi}\sigma' \bigl( \Vert \nabla u_{1} \Vert ^{2}\bigr) \int_{\Omega}\nabla u_{1}\nabla u_{1t}\,dx \Vert \nabla w \Vert ^{2}\,d\xi \leq C_{t_{0},\tau} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert \nabla w \Vert ^{2}\,d\xi . \end{aligned}$$
(5.27)
By the value theorem and (4.19) we have
$$\begin{aligned} \begin{aligned}[b] - \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \int_{\Omega}\triangle\sigma w_{t}\,dx\,d\xi &\leq C_{t_{0},\tau} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w \Vert \Vert w_{t} \Vert \,d\xi \\ &\leq \frac{C_{t_{0},\tau}^{2}}{4\varepsilon'} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w \Vert ^{2}\,d\xi +\varepsilon' \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w_{t} \Vert ^{2}\,d\xi . \end{aligned} \end{aligned}$$
(5.28)
From (5.26) combined with (5.27)-(5.28) we have
$$\begin{aligned} & \bigl(2+5\varepsilon'\bigr) \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta} \xi} \Vert w_{t} \Vert ^{2}\,d\xi \\ &\quad \leq \frac{(2+5\varepsilon')}{\eta-\varepsilon'}e^{\tilde{\delta}(t_{0}-\tau)}F(t_{0}-\tau) + \frac{(2+5\varepsilon')\tilde{\delta}}{\eta-\varepsilon'} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi}F(\xi)\,d\xi \\ &\qquad {}-\frac{(2+5\varepsilon')}{\eta-\varepsilon'} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \int_{\Omega}\triangle g w_{t}\,dx\,d\xi + \frac{(2+5\varepsilon')}{\eta-\varepsilon'}C_{t_{0},\tau} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert \nabla w \Vert ^{2} \,d\xi \\ &\qquad {} +\frac{(2+5\varepsilon')}{\eta-\varepsilon'}\frac{C^{2}_{t_{0},\tau}}{4\varepsilon'} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w \Vert ^{2} \,d\xi -\frac{(2+5\varepsilon')}{\eta-\varepsilon'} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert \nabla \tilde{ \theta} \Vert ^{2}\,d\xi . \end{aligned}$$
(5.29)
Substituting (5.29) into (5.25), we obtain
$$\begin{aligned} &\tau e^{\tilde{\delta}t_{0}}F(t_{0})+ \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}s}F(s)\,ds \\ &\quad \leq \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert \tilde{\theta} \Vert ^{2}\,d\xi -\biggl(\frac{1}{2}\tilde{\delta}\tau+1 \biggr) e^{\tilde{\delta}t_{0}}\bigl(w_{t}(t_{0}),w(t_{0}) \bigr) + C^{2}_{t_{0},\tau} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w \Vert ^{2}\,d\xi \\ &\qquad {}+\frac{(2+5\varepsilon')}{\eta-\varepsilon'}e^{\tilde{\delta}(t_{0}-\tau)}F(t_{0}-\tau) + \frac{(2+5\varepsilon')\tilde{\delta}}{\eta-\varepsilon'} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi}F(\xi)\,d\xi \\ &\qquad {}-\frac{(2+5\varepsilon')}{\eta-\varepsilon'} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \int_{\Omega}\triangle g w_{t}\,dx\,d\xi - \frac{(2+5\varepsilon')}{\eta-\varepsilon'} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert \nabla \tilde{ \theta} \Vert ^{2}\,d\xi \\ &\qquad {}+2C_{t_{0},\tau}\biggl( \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w \Vert ^{2}\,d\xi \biggr)^{\frac{1}{2}}+\frac{2\varepsilon'}{\lambda_{0}^{2}} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert \nabla\tilde{ \theta} \Vert ^{2}\,d\xi +C^{3}_{t_{0},\tau} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert \nabla w \Vert ^{2}\,d\xi \\ &\qquad {}+ \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi} \int_{\Omega}\triangle g w_{t}\,dx\,d\xi \,ds +e^{\tilde{\delta}(t_{0}-\tau)}\bigl(w_{t}(t_{0}-\tau),w(t_{0}- \tau)\bigr), \end{aligned}$$
(5.30)
where \(C^{2}_{t_{0},\tau}=\frac{\tilde{\delta}^{2}}{8\varepsilon'}+\frac{[\frac{1}{2}\tau\tilde{\delta} (\tilde{\delta}-\eta)]^{2}}{4\varepsilon'}+\frac{(\tau C_{t_{0},\tau})^{2}}{4\varepsilon'}+\frac{(\tilde{\delta}-\eta)^{2}}{8\varepsilon'}+2C_{t_{0},\tau}+\frac{(2+5\varepsilon')}{\eta-\varepsilon'}\frac{C^{2}_{t_{0},\tau}}{4\varepsilon'}\) and \(C^{3}_{t_{0},\tau}=\frac{(\tau \frac{1}{2}\gamma\tilde{\delta})^{2}}{4\varepsilon'}+\frac{\lambda_{0}^{2}\gamma^{2}}{4\varepsilon'}+C \tau C_{t_{0},\tau}+\tau C_{t_{0},\tau}+\frac{(2+5\varepsilon')}{\eta-\varepsilon'}C_{t_{0},\tau}\).
Since \(0<\varepsilon'<\min\{\frac{\sqrt{(9+\lambda_{0}^{2})^{2}+40\eta\lambda_{0}^{2}}-(9+\lambda_{0}^{2})}{20},\eta\}\), we have
$$-\frac{(2+5\varepsilon')}{\eta-\varepsilon'} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert \nabla \tilde{ \theta} \Vert ^{2}\,d\xi +\frac{2\varepsilon'}{\lambda_{0}^{2}} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert \nabla\tilde{ \theta} \Vert ^{2}\,d\xi + \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert \tilde{\theta} \Vert ^{2}\,d\xi \leq0. $$
So from (5.30) we obtain
$$\begin{aligned} \begin{aligned}[b] &\tau e^{\tilde{\delta}t_{0}}F(t_{0})+ \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}s}F(s)\,ds \\ &\quad \leq -\biggl(\frac{1}{2}\tilde{\delta}\tau+1\biggr) e^{\tilde{\delta}t_{0}} \bigl(w_{t}(t_{0}),w(t_{0})\bigr) +C^{2}_{t_{0},\tau} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w \Vert ^{2}\,d\xi \\ &\qquad {}+\frac{(2+5\varepsilon')}{\eta-\varepsilon'}e^{\tilde{\delta}(t_{0}-\tau)}F(t_{0}-\tau) + \frac{(2+5\varepsilon')\tilde{\delta}}{\eta-\varepsilon'} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi}F(\xi)\,d\xi \\ &\qquad {}-\frac{2(2+5\varepsilon')}{\eta-\varepsilon'} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \int_{\Omega}\triangle g w_{t}\,dx\,d\xi \\ &\qquad {}+2C_{t_{0},\tau}\biggl( \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w \Vert ^{2}\,d\xi \biggr)^{\frac{1}{2}}+C^{3}_{t_{0},\tau} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert \nabla w \Vert ^{2}\,d\xi \\ &\qquad {}+ \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi} \int_{\Omega}\triangle g w_{t}\,dx\,d\xi \,ds +e^{\tilde{\delta}(t_{0}-\tau)}\bigl(w_{t}(t_{0}-\tau),w(t_{0}- \tau)\bigr). \end{aligned} \end{aligned}$$
(5.31)
Since \(\tilde{\delta}\leq\frac{\eta-\varepsilon'}{2+5\varepsilon'}\), we have
$$\begin{aligned} \begin{aligned}[b] \tau e^{\tilde{\delta}t_{0}}F(t_{0}) \leq{} &{-}\biggl(\frac{1}{2}\tilde{\delta}\tau+1\biggr) e^{\tilde{\delta}t_{0}} \bigl(w_{t}(t_{0}),w(t_{0})\bigr) +C^{2}_{t_{0},\tau} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w \Vert ^{2}\,d\xi \\ &{}+\frac{(2+5\varepsilon')}{\eta-\varepsilon'}e^{\tilde{\delta}(t_{0}-\tau)}F(t_{0}-\tau) - \frac{(2+5\varepsilon')}{\eta-\varepsilon'} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \int_{\Omega}\triangle g w_{t}\,dx\,d\xi \\ &{}+2C_{t_{0},\tau}\biggl( \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w \Vert ^{2}\,d\xi \biggr)^{\frac{1}{2}}+C^{3}_{t_{0},\tau} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert \nabla w \Vert ^{2}\,d\xi \\ &{}+ \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi} \int_{\Omega}\triangle g w_{t}\,dx\,d\xi \,ds +e^{\tilde{\delta}(t_{0}-\tau)}\bigl(w_{t}(t_{0}-\tau),w(t_{0}- \tau)\bigr). \end{aligned} \end{aligned}$$
(5.32)
By (5.3) we have
$$\begin{aligned} \begin{aligned}[b] E_{w}(t_{0}) \leq{}&{-} \frac{1}{C^{1}}\biggl(\frac{1}{2}\tilde{\delta}+\frac{1}{\tau} \biggr) \bigl(w_{t}(t_{0}),w(t_{0})\bigr) + \frac{1}{C^{1}\tau} C^{2}_{t_{0},\tau}e^{-\tilde{\delta}t_{0}} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w \Vert ^{2}\,d\xi \\ &{}+\frac{(2+5\varepsilon')}{C^{1}(\eta-\varepsilon')\tau}e^{\tilde{\delta}(-\tau)}F(t_{0}-\tau) - \frac{(2+5\varepsilon')}{C^{1}(\eta-\varepsilon')\tau}e^{-\tilde{\delta}t_{0}} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \int_{\Omega}\triangle g w_{t}\,dx\,d\xi \\ &{} +2C_{t_{0},\tau}\biggl( \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w \Vert ^{2}\,d\xi \biggr)^{\frac{1}{2}}+\frac{1}{C^{1}\tau}C^{3}_{t_{0},\tau} e^{-\tilde{\delta}t_{0}} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert \nabla w \Vert ^{2}\,d\xi \\ &{}+\frac{1}{C^{1}\tau}e^{-\tilde{\delta}t_{0}} \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi} \int_{\Omega}\triangle g w_{t}\,dx\,d\xi \,ds + \frac{1}{C^{1}\tau}e^{-\tilde{\delta}\tau}\bigl(w_{t}(t_{0}- \tau),w(t_{0}-\tau)\bigr). \end{aligned} \end{aligned}$$
(5.33)
We set
$$\begin{aligned} \begin{aligned}[b] &\phi_{t_{0},\tau}\bigl(\bigl(u_{0}^{1},v_{0}^{1}, \theta_{0}^{1}\bigr),\bigl(u_{0}^{2},v_{0}^{2}, \theta_{0}^{2}\bigr)\bigr) \\ &\quad =-\frac{1}{C^{1}}\biggl(\frac{1}{2}\tilde{\delta}+ \frac{1}{\tau}\biggr) \bigl(w_{t}(t_{0}),w(t_{0}) \bigr) +\frac{1}{C^{1}\tau} e^{-\tilde{\delta}t_{0}}C^{2}_{t_{0},\tau} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w \Vert ^{2}\,d\xi \\ &\qquad {}-\frac{(2+5\varepsilon')}{C^{1}\lambda^{2}\tau}e^{-\tilde{\delta}t_{0}} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \int_{\Omega}\triangle g w_{t}\,dx\,d\xi +2C_{t_{0},\tau}\biggl( \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert w \Vert ^{2}\,d\xi \biggr)^{\frac{1}{2}} \\ &\qquad {}+\frac{1}{C^{1}\tau}C^{3}_{t_{0},\tau} e^{-\tilde{\delta}t_{0}} \int_{t_{0}-\tau}^{t_{0}}e^{\tilde{\delta}\xi} \Vert \nabla w \Vert ^{2}\,d\xi +\frac{1}{C^{1}\tau}e^{-\tilde{\delta}t_{0}} \int_{t_{0}-\tau}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi} \int_{\Omega}\triangle g w_{t}\,dx\,d\xi \,ds. \end{aligned} \end{aligned}$$
(5.34)
Since \(\lim_{\tau\rightarrow\infty}e^{-\hat{\sigma}\tau}\tilde{R}^{2}_{t_{0}-\tau}=0\), for any \(\varepsilon>0\), we can find \(\tau_{0}=\tau_{0}(\varepsilon,\tilde{D},t_{0})\geq 0\) such that
$$\frac{(2+5\varepsilon')}{C^{1}\lambda^{2}\tau}e^{\tilde{\delta}(-\tau)}F(t_{0}-\tau) +\frac{1}{C^{1}\tau}e^{-\tilde{\delta}\tau} \bigl(w_{t}(t_{0}-\tau),w(t_{0}-\tau)\bigr)\leq \varepsilon. $$
Thus we have
$$E_{w}(t_{0})\leq\varepsilon+\phi_{t_{0},\tau}\bigl( \bigl(u_{0}^{1},v_{0}^{1}, \theta_{0}^{1}\bigr),\bigl(u_{0}^{2},v_{0}^{2}, \theta_{0}^{2}\bigr)\bigr)\quad \mbox{for all } \bigl(u_{0}^{i},v_{0}^{i}, \theta_{0}^{i}\bigr)\in\tilde{D}_{t_{0}-\tau_{0}}. $$
By Lemma 3.1 we only need to show that \(\phi_{t_{0},\tau_{0}}(\cdot,\cdot)\) defined by (5.34) is a contractive function on \(\tilde{D}_{t_{0}-\tau_{0}} \times \tilde{D}_{t_{0}-\tau_{0}}\). Let \((u_{n},u_{nt},\theta_{n})\) be the corresponding solutions of \((u_{0}^{n}, v_{0}^{n}, \theta_{0}^{n})\in\tilde{D}_{t_{0}-\tau_{0}}, n=1,2,\dots\). Since \(\tilde{D}_{t_{0}-\tau_{0}}\) is a bounded subset in \(X_{0}\), by (4.16) we know that
$$\begin{aligned} \bigl\Vert \bigl(u_{n}(s),u_{nt}(s), \theta_{n}(s)\bigr) \bigr\Vert _{X_{0}}\leq C'_{t_{0},\tau_{0}}< +\infty\quad \mbox{for all } s\in[t_{0}- \tau_{0},t_{0}] \mbox{ and } n\in N, \end{aligned}$$
(5.35)
where \(C'_{t_{0},\tau_{0}}\) depends on \(t_{0},\tau_{0}\).
Now, we will deal with the right terms in (5.34) one by one.
Without loss of generality, assuming first that
$$u_{n}\rightarrow u\quad \mbox{weak-star in } L^{\infty} \bigl(t_{0}-\tau_{0}, t_{0}; H_{0}^{2}( \Omega)\bigr) $$
and considering that compact embeddings \(H_{0}^{2}(\Omega)\hookrightarrow\hookrightarrow H_{0}^{1}(\Omega) \), we have
$$u_{n}\rightarrow u\quad \mbox{strongly in } L^{2} \bigl(t_{0}-\tau_{0}, t_{0}; H_{0}^{1}( \Omega)\bigr), $$
so we obtain
$$\begin{aligned} \lim_{n\rightarrow\infty}\lim_{m\rightarrow\infty} \int_{t_{0}-\tau_{0}}^{t_{0}} e^{\tilde{\delta}\xi} \bigl\Vert \nabla u_{n}(\xi)-\nabla u_{m}(\xi) \bigr\Vert ^{2}\,d \xi =0. \end{aligned}$$
(5.36)
Second, similarly assuming that
$$u_{n}\rightarrow u\quad \mbox{weak-star in } L^{\infty} \bigl(t_{0}-\tau_{0}, t_{0}; H_{0}^{1}( \Omega)\bigr) $$
and considering compact embeddings \(H_{0}^{1}(\Omega)\hookrightarrow \hookrightarrow L^{2}(\Omega)\), we also get
$$\begin{aligned} \begin{aligned} &\lim_{n\rightarrow\infty}\lim _{m\rightarrow\infty} \int_{t_{0}-\tau_{0}}^{t_{0}}e^{\tilde{\delta}\xi} \bigl\Vert u_{n}(\xi)- u_{m}(\xi) \bigr\Vert ^{2}\,d\xi =0, \\ &\lim_{n\rightarrow\infty}\lim_{m\rightarrow\infty}\biggl( \int_{t_{0}-\tau_{0}}^{t_{0}}e^{\tilde{\delta}\xi} \bigl\Vert u_{n}(\xi)- u_{m}(\xi) \bigr\Vert ^{2}\,d\xi \biggr)^{\frac{1}{2}}=0. \end{aligned} \end{aligned}$$
(5.37)
Finally, with
$$\begin{aligned} &u_{n}\rightarrow u\quad \mbox{weak-star in } L^{\infty} \bigl(t_{0}-\tau_{0}, t_{0}; L^{2}(\Omega) \bigr), \\ &u_{nt}\rightarrow u_{t}\quad \mbox{weak-star in } L^{\infty}\bigl(t_{0}-\tau_{0}, t_{0}; L^{2}(\Omega)\bigr), \end{aligned}$$
we have
$$u_{n}\rightarrow u\quad \mbox{strongly in } C\bigl(t_{0}- \tau_{0}, t_{0}; L^{2}(\Omega)\bigr). $$
So, it is easy to obtain that
$$\begin{aligned} \begin{aligned}[b] &\lim_{n\rightarrow\infty}\lim _{m\rightarrow\infty} \int_{\Omega} \bigl(u_{nt}(t_{0})-u_{mt}(t_{0}) \bigr) \bigl(u_{n}(t_{0})-u_{m}(t_{0}) \bigr)\,dx \\ &\quad =\lim_{n\rightarrow\infty}\lim_{m\rightarrow\infty} \int_{\Omega} \bigl(u_{nt}(t_{0})-u_{mt}(t_{0}) \bigr) \bigl(u_{n}(t_{0})-u(t_{0})+u(t_{0})-u_{m}(t_{0}) \bigr)\,dx \\ &\quad =\lim_{n\rightarrow\infty}\lim_{m\rightarrow\infty} \int_{\Omega} \bigl(u_{nt}(t_{0})-u_{mt}(t_{0}) \bigr) \bigl(u_{n}(t_{0})-u(t_{0})\bigr)\,dx \\ &\qquad {}+\lim_{n\rightarrow\infty}\lim_{m\rightarrow\infty} \int_{\Omega} \bigl(u_{nt}(t_{0})-u_{mt}(t_{0}) \bigr) \bigl(u(t_{0})-u_{m}(t_{0})\bigr)\,dx \\ &\quad \leq \bigl\Vert u_{nt}(t_{0})-u_{mt}(t_{0}) \bigr\Vert _{L^{\infty}(L^{2}(\Omega))} \bigl\Vert u_{n}(t_{0})-u(t_{0}) \bigr\Vert \\ &\qquad {}+ \bigl\Vert u_{nt}(t_{0})-u_{mt}(t_{0}) \bigr\Vert _{L^{\infty}(L^{2}(\Omega))} \bigl\Vert u(t_{0})-u_{m}(t_{0}) \bigr\Vert \\ &\quad \leq C\bigl[ \bigl\Vert u_{n}(t_{0})-u(t_{0}) \bigr\Vert + \bigl\Vert u(t_{0})-u_{m}(t_{0}) \bigr\Vert \bigr] \\ &\quad \rightarrow 0. \end{aligned} \end{aligned}$$
(5.38)
By assumptions (1.6) on \(g(\cdot)\), using the embedding theorem combined with (5.35), we have
$$\begin{aligned} \begin{aligned}[b] &\lim_{n\rightarrow\infty}\lim _{m\rightarrow\infty} \int_{t_{0}-\tau_{0}}^{t_{0}} e^{\tilde{\delta}\xi} \int_{\Omega}\bigl(u_{nt}(\xi)-u_{mt}(\xi) \bigr) \bigl(g\bigl(u_{n}(\xi)\bigr)-g\bigl(u_{m}(\xi)\bigr) \bigr)\,dx\,d\xi \\ &\quad \leq \lim_{n\rightarrow\infty}\lim_{m\rightarrow\infty} \int_{t_{0}-\tau_{0}}^{t_{0}} e^{\tilde{\delta}\xi} \int_{\Omega}\bigl(u_{nt}(\xi)-u_{mt}(\xi) \bigr)k_{2}\bigl(u_{n}(\xi)-u_{m}(\xi)\bigr)\\ &\qquad {}\times \bigl(1+ \bigl\vert u_{n}(\xi) \bigr\vert ^{\rho-1}+ \bigl\vert u_{m}(\xi) \bigr\vert ^{\rho-1}\bigr)\,dx\,d\xi \\ &\quad \leq C'_{t_{0},\tau_{0}} \lim_{n\rightarrow\infty}\lim _{m\rightarrow\infty}\biggl( \int_{t_{0}-\tau_{0}}^{t_{0}} e^{\tilde{\delta}\xi} \bigl\Vert \bigl(u_{n}(\xi)-u_{m}(\xi)\bigr) \bigr\Vert ^{2}\,d\xi \biggr)^{\frac{1}{2}} \\ &\quad =0. \end{aligned} \end{aligned}$$
(5.39)
Similarly, since \(\int_{s}^{t_{0}}e^{\tilde{\delta}\xi}\int_{\Omega} (u_{nt}(\xi)-u_{mt}(\xi))(g(u_{n}(\xi))-g(u_{m}(\xi)))\,dx\,d\xi \) is bounded for each \(s\in[\tau,t_{0}]\), by (5.39) and the Lebesgue dominated convergence theorem we have
$$\begin{aligned} &\lim_{n\rightarrow\infty}\lim _{m\rightarrow\infty} \int_{t_{0}-\tau_{0}}^{t_{0}} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi} \int_{\Omega} \bigl(u_{nt}(\xi)-u_{mt}(\xi) \bigr) \bigl(g\bigl(u_{n}(\xi)\bigr)-g\bigl(u_{m}(\xi)\bigr) \bigr)\,dx\,d\xi \,ds \\ &\quad = \int_{t_{0}-\tau_{0}}^{t_{0}}\biggl(\lim_{n\rightarrow\infty}\lim _{m\rightarrow\infty} \int_{s}^{t_{0}}e^{\tilde{\delta}\xi} \int_{\Omega} \bigl(u_{nt}(\xi)-u_{mt}(\xi) \bigr) \bigl(g\bigl(u_{n}(\xi)\bigr)-g\bigl(u_{m}(\xi)\bigr) \bigr)\,dx\,d\xi \biggr)\,ds \\ &\quad = \int_{t_{0}-\tau_{0}}^{t_{0}}0\,ds \\ &\quad =0. \end{aligned}$$
(5.40)
Combining (5.36)-(5.40), we get that \(\Phi_{t_{0},\tau_{0}}(\cdot,\cdot)\) is a contractive function on \(\tilde{D}_{t_{0}-\tau_{0}}\times\tilde{D}_{t_{0}-\tau_{0}}\). The proof is finished by Lemma 3.1. □