In this section, we derive uniform estimates on the solutions of the stochastic plate equations (3.2)–(3.3) defined on \(\mathbb{R}^{n}\).
We define a new norm \(\|\cdot\|_{E}\) by
$$ \Vert Y \Vert _{E}=\bigl( \Vert v \Vert ^{2}+ \bigl(\lambda+\delta^{2}-\beta_{2}\delta\bigr) \Vert u \Vert ^{2}+ \Vert \Delta u \Vert ^{2} \bigr)^{\frac{1}{2}}\quad \text{for } Y=(u,v)\in E. $$
(4.1)
It is easy to check that \(\|\cdot\|_{E}\) is equivalent to the usual norm \(\|\cdot\|_{H^{2}\times L^{2}}\) in (3.1).
The next lemma shows that the cocycle Φ has a pullback \(\mathcal{D}\)-absorbing set in \(\mathcal{D}\).
Lemma 4.1
Under (3.4)–(3.9) and (3.14)–(3.17), for every\(\tau\in\mathbb{R}\), \(\omega\in\varOmega\), \(D=\{D(\tau,\omega):\tau\in\mathbb{R},\omega\in\varOmega\}\in \mathcal{D}\), there exists\(T=T(\tau,\omega,D)>0\)such that, for all\(t\geq T\), the solution of problem (3.10)–(3.11) satisfies
$$\begin{aligned}& \bigl\Vert Y\bigl(\tau,\tau-t,\theta_{-\tau}\omega,D(\tau-t, \theta_{-t} \omega)\bigr) \bigr\Vert ^{2}_{E} \leq R_{1}(\tau,\omega), \\& e^{-\sigma t} \int^{t}_{\tau-t}e^{\sigma s} \bigl\Vert Y \bigl(s,\tau-t,\theta_{- \tau}\omega,D(\tau-t,\theta_{-t} \omega)\bigr) \bigr\Vert ^{2}_{E}\,ds\leq R_{1}( \tau,\omega), \end{aligned}$$
and\(R_{1}(\tau,\omega)\)is given by
$$ \begin{aligned}R_{1}(\tau,\omega)=M+M \int^{\tau}_{-\infty }e^{\sigma(s- \tau)} \bigl\Vert g(x, s) \bigr\Vert ^{2}\,ds+c \int^{0}_{ -\infty}e^{\sigma s }\bigl( \bigl\vert \omega(s) \bigr\vert ^{2}+ \bigl\vert \omega(s) \bigr\vert ^{p+1}\bigr)\,ds, \end{aligned} $$
(4.2)
whereMis a positive constant independent ofτ, ω, D.
Proof
Taking the inner product of the second equation of (3.10) with v in \(L^{2}(\mathbb{R}^{n})\), we find that
$$\begin{aligned} &\frac{1}{2}\frac{d}{dt} \Vert v \Vert ^{2}- \delta \Vert v \Vert ^{2}+\bigl(\lambda+\delta^{2} \bigr) (u,v)+ (Au,v)+\bigl(f(x,u),v\bigr) \\ &\quad =\bigl(g(x,t),v\bigr)-\bigl(h\bigl(v+ \phi\omega(t)-\delta u\bigr),v \bigr)+ \delta(\phi,v) \omega(t). \end{aligned}$$
(4.3)
By the first equation of (3.10), we have
$$ v=u_{t}- \phi\omega(t)+\delta u. $$
(4.4)
By Lagrange’s mean value theorem and (3.9), we get
$$\begin{aligned}& -\bigl(h\bigl(v+ \phi\omega(t)-\delta u\bigr),v\bigr) \\& \quad = -\bigl(h\bigl(v+ \phi\omega(t)-\delta u\bigr)-h(0),v\bigr) \\& \quad = -\bigl(h'(\vartheta) \bigl(v+ \phi\omega(t)-\delta u \bigr),v\bigr) \\& \quad \leq -\beta_{1} \Vert v \Vert ^{2}- \bigl(h'(\vartheta) \bigl( \phi\omega(t)-\delta u\bigr),v\bigr) \\& \quad \leq -\beta_{1} \Vert v \Vert ^{2}+ \beta_{2} \bigl\vert \omega(t) \bigr\vert \Vert \phi \Vert \Vert v \Vert +h'( \vartheta)\delta(u,v) \\& \quad \leq -\beta_{1} \Vert v \Vert ^{2}+ \frac{\beta_{1}-\delta}{6} \Vert v \Vert ^{2}+ \frac{3\beta_{2}^{2}}{2(\beta_{1}-\delta)} \bigl\vert \omega(t) \bigr\vert ^{2} \Vert \phi \Vert ^{2}+h'(\vartheta)\delta(u,v), \end{aligned}$$
(4.5)
where ϑ is between 0 and \(v+ \phi\omega(t)-\delta u\).
By (3.9) and (4.4), we get
$$\begin{aligned}& h'(\vartheta)\delta(u,v) \\& \quad = h'(\vartheta)\delta\bigl(u,u_{t}-\phi \omega(t)+\delta u\bigr) \\& \quad \leq \beta_{2}\delta\cdot\frac{1}{2} \frac{d}{dt} \Vert u \Vert ^{2}+\beta_{2} \delta^{2} \Vert u \Vert ^{2}+ \beta_{2} \delta \bigl\vert \omega(t) \bigr\vert \Vert \phi \Vert \Vert u \Vert \\& \quad \leq \beta_{2}\delta\cdot\frac{1}{2} \frac{d}{dt} \Vert u \Vert ^{2}+\beta_{2} \delta^{2} \Vert u \Vert ^{2}+\frac{1}{4} \delta\bigl(\lambda+\delta^{2}-\beta_{2} \delta\bigr) \Vert u \Vert ^{2} +c \bigl\vert \omega(t) \bigr\vert ^{2} \Vert \phi \Vert ^{2}. \end{aligned}$$
(4.6)
Then substituting the v in (4.4) into the third and fourth terms on the left-hand side of (4.3), we find that
$$\begin{aligned}& \bigl(\lambda+\delta^{2}\bigr) (u,v) \\& \quad =\bigl(\lambda+\delta^{2}\bigr) \bigl(u,u_{t}- \phi\omega(t)+\delta u\bigr) \\& \quad \geq\frac{1}{2}\bigl(\lambda+\delta^{2}\bigr) \frac{d}{dt} \Vert u \Vert ^{2}+\delta\bigl( \lambda+ \delta^{2}\bigr) \Vert u \Vert ^{2}-\bigl(\lambda+ \delta^{2}\bigr) \bigl\vert \omega(t) \bigr\vert \Vert \phi \Vert \Vert u \Vert \\& \quad \geq\frac{1}{2}\bigl(\lambda+\delta^{2}\bigr) \frac{d}{dt} \Vert u \Vert ^{2}+\delta\bigl( \lambda+ \delta^{2}\bigr) \Vert u \Vert ^{2}- \frac{1}{4}\delta\bigl(\lambda+\delta^{2}- \beta_{2}\delta\bigr) \Vert u \Vert ^{2} - c \bigl\vert \omega(t) \bigr\vert ^{2} \Vert \phi \Vert ^{2}, \end{aligned}$$
(4.7)
$$\begin{aligned}& \begin{aligned}[b] (Au,v) &=(\Delta u,\Delta v)=\bigl(\Delta u,\Delta u_{t}-\omega(t)\Delta \phi+\delta\Delta u\bigr) \\ &\geq\frac{1}{2}\frac{d}{dt} \Vert \Delta u \Vert ^{2}+\delta \Vert \Delta u \Vert ^{2}- \bigl\vert \omega(t) \bigr\vert \Vert \Delta\phi \Vert \Vert \Delta u \Vert \\ &\geq\frac{1}{2}\frac{d}{dt} \Vert \Delta u \Vert ^{2}+\frac{\delta}{2} \Vert \Delta u \Vert ^{2}- \frac{1}{2\delta} \bigl\vert \omega(t) \bigr\vert ^{2} \Vert \Delta\phi \Vert ^{2}. \end{aligned} \end{aligned}$$
(4.8)
Using the Cauchy–Schwarz inequality and Young’s inequality, we have
$$ \begin{aligned}\delta\bigl(\phi\omega(t),v\bigr)\leq\delta \bigl\vert \omega(t) \bigr\vert \Vert \phi \Vert \Vert v \Vert \leq \frac{3 \delta^{2}}{2(\beta_{1}-\delta)} \Vert \phi \Vert ^{2} \bigl\vert \omega(t) \bigr\vert ^{2}+\frac{\beta_{1}-\delta}{6} \Vert v \Vert ^{2} \end{aligned} $$
(4.9)
and
$$ (g,v)\leq \Vert g \Vert \Vert v \Vert \leq\frac{3}{2(\beta_{1}-\delta)} \Vert g \Vert ^{2}+ \frac{\beta_{1}-\delta}{6} \Vert v \Vert ^{2}. $$
(4.10)
Let \(\widetilde{F}(x,u)=\int_{\mathbb{R}^{n}}F(x,u)\,dx\). Then, for the last term on the left-hand side of (4.3), we have
$$ \begin{aligned}\bigl(f(x,u),v\bigr)&=\bigl(f(x,u),u_{t}- \phi\omega(t) +\delta u\bigr) \\ &=\frac{d}{dt}\widetilde{F}(x,u)+\delta\bigl(f(x,u),u\bigr)- \bigl(f(x,u),\phi \omega(t)\bigr). \end{aligned} $$
(4.11)
By condition (3.5) we get
$$ \bigl(f(x,u),u\bigr)\geq c_{2}\widetilde{F}(x,u)+ \int_{\mathbb{R}^{n}}\eta_{2}(x)\,dx. $$
(4.12)
Using condition (3.4) and (3.6), we obtain
$$\begin{aligned}& \bigl(f(x,u),\phi\omega(t)\bigr) \\& \quad \leq \int_{\mathbb{R}^{n}}\bigl(c_{1} \vert u \vert ^{p}+\eta_{1}(x)\bigr) \bigl\vert \phi\omega(t) \bigr\vert \,dx \\& \quad \leq \bigl\Vert \eta_{1}(x) \bigr\Vert \Vert \phi \Vert \bigl\vert \omega(t) \bigr\vert +c_{1}\biggl( \int_{\mathbb {R}^{n}} \vert u \vert ^{p+1}\,dx \biggr)^{ \frac{p}{p+1}} \Vert \phi \Vert _{p+1} \bigl\vert \omega(t) \bigr\vert \\& \quad \leq \bigl\Vert \eta_{1}(x) \bigr\Vert \Vert \phi \Vert \bigl\vert \omega(t) \bigr\vert +c_{1}\biggl( \int_{\mathbb{R}^{n}}\bigl(F(x,u)+ \eta_{3}(x)\bigr)\,dx \biggr)^{\frac{p}{p+1}} \Vert \phi \Vert _{p+1} \bigl\vert \omega(t) \bigr\vert \\& \quad \leq \frac{1}{2} \bigl\Vert \eta_{1}(x) \bigr\Vert ^{2}+\frac{1}{2} \Vert \phi \Vert ^{2} \bigl\vert \omega(t) \bigr\vert ^{2}+ \frac{\delta c_{2}}{2}\widetilde{F}(x,u) \\& \qquad {}+ \frac{\delta c_{2}}{2} \int_{\mathbb{R}^{n}}\eta_{3}(x)\,dx+c \Vert \phi \Vert ^{p+1}_{H^{2}} \bigl\vert \omega(t) \bigr\vert ^{p+1}. \end{aligned}$$
(4.13)
By (4.11)–(4.13), we get
$$\begin{aligned}& \delta\bigl(f(x,u),u\bigr)-\bigl(f(x,u),\phi\omega(t) \bigr) \\& \quad \geq \frac{\delta c_{2}}{2}\widetilde{F}(x,u)+\delta \int_{ \mathbb{R}^{n}}\eta_{2}(x)\,dx-\frac{1}{2} \bigl\Vert \eta_{1}(x) \bigr\Vert ^{2}- \frac{1}{2} \Vert \phi \Vert ^{2} \bigl\vert \omega(t) \bigr\vert ^{2} \\& \qquad {} -\frac{\delta c_{2}}{2} \int_{\mathbb{R}^{n}}\eta_{3}(x)\,dx-c \Vert \phi \Vert ^{p+1}_{H^{2}} \bigl\vert \omega(t) \bigr\vert ^{p+1}. \end{aligned}$$
(4.14)
Substitute (4.5)–(4.14) into (4.3) to obtain
$$\begin{aligned}& \frac{1}{2}\frac{d}{dt}\bigl( \Vert v \Vert ^{2}+\bigl(\lambda+\delta^{2}-\beta_{2} \delta\bigr) \Vert u \Vert ^{2}+ \Vert \Delta u \Vert ^{2}+2\widetilde{F}(x,u)\bigr) \\& \qquad {} +\delta\bigl( \Vert v \Vert ^{2}+\bigl(\lambda+ \delta^{2}-\beta_{2}\delta\bigr) \Vert u \Vert ^{2}+ \Vert \Delta u \Vert ^{2}\bigr)+ \frac{\delta c_{2}}{2}\widetilde{F}(x,u) \\& \quad \leq \frac{\delta}{2} \Vert v \Vert ^{2}+ \frac{2\delta-\beta_{1} }{2} \Vert v \Vert ^{2}+ \frac{\delta}{2} \bigl(\lambda+\delta^{2}-\beta_{2}\delta\bigr) \Vert u \Vert ^{2} + \frac{\delta}{2} \Vert \Delta u \Vert ^{2} \\& \qquad {} +c\bigl(1+ \bigl\vert \omega(t) \bigr\vert ^{2}+ \bigl\vert \omega(t) \bigr\vert ^{p+1}\bigr)+ \frac{3}{2(\beta_{1}-\delta)} \Vert g \Vert ^{2}. \end{aligned}$$
(4.15)
Let \(\sigma=\min\{\delta,\frac{\delta c_{2}}{2}\}\), then
$$\begin{aligned}& \frac{d}{dt}\bigl( \Vert v \Vert ^{2}+\bigl(\lambda+ \delta^{2}-\beta_{2}\delta\bigr) \Vert u \Vert ^{2}+ \Vert \Delta u \Vert ^{2}+2\widetilde{F}(x,u) \bigr) \\& \qquad {} +\sigma\bigl( \Vert v \Vert ^{2}+\bigl(\lambda+ \delta^{2}-\beta_{2}\delta\bigr) \Vert u \Vert ^{2}+ \Vert \Delta u \Vert ^{2}+2\widetilde{F}(x,u) \bigr) \\& \quad \leq \frac{3}{ (\beta_{1}-\delta)} \Vert g \Vert ^{2}+c\bigl(1+ \bigl\vert \omega(t) \bigr\vert ^{2}+ \bigl\vert \omega(t) \bigr\vert ^{p+1}\bigr). \end{aligned}$$
(4.16)
Multiplying (4.16) by \(e^{\sigma t}\) and then integrating over \((\tau-t,\tau)\), we have
$$\begin{aligned}& e^{\sigma\tau} \bigl( \bigl\Vert v(\tau,\tau-t,\omega,v_{0}) \bigr\Vert ^{2}+\bigl(\lambda+ \delta^{2}- \beta_{2}\delta\bigr) \bigl\Vert u(\tau,\tau-t,\omega,u_{0}) \bigr\Vert ^{2} \\& \qquad {} + \bigl\Vert \Delta u(\tau,\tau-t,\omega,u_{0}) \bigr\Vert ^{2}+2\widetilde{F}\bigl(x,( \tau,\tau-t,\omega,u_{0}) \bigr)\bigr) \\& \quad \leq e^{\sigma(\tau-t)}\bigl( \Vert v_{0} \Vert ^{2}+\bigl(\lambda+\delta^{2}-\beta_{2} \delta\bigr) \Vert u_{0} \Vert ^{2}+ \Vert \Delta u_{0} \Vert ^{2}+2\widetilde{F}(x,u_{0}) \bigr) \\& \qquad {} +\frac{3}{ (\beta_{1}-\delta)} \int^{\tau}_{\tau-t}e^{\sigma s} \bigl\Vert g(x,s) \bigr\Vert ^{2}\,ds+c \int^{\tau}_{\tau-t}e^{\sigma s}\bigl(1+ \bigl\vert \omega(s) \bigr\vert ^{2}+ \bigl\vert \omega(s) \bigr\vert ^{p+1}\bigr)\,ds. \end{aligned}$$
Replacing ω by \(\theta_{-\tau}\omega\) in the above, we obtain, for every \(t\in\mathbb{R}^{+}\), \(\tau\in\mathbb{R}\), and \(\omega\in \varOmega\),
$$\begin{aligned}& \bigl\Vert v(\tau,\tau-t,\theta_{-\tau}\omega,v_{0}) \bigr\Vert ^{2}+\bigl(\lambda+ \delta^{2}- \beta_{2}\delta\bigr) \bigl\Vert u(\tau,\tau-t,\theta_{-\tau} \omega,u_{0}) \bigr\Vert ^{2} \\& \qquad {} + \bigl\Vert \Delta u(\tau,\tau-t,\omega,u_{0}) \bigr\Vert ^{2}+2\widetilde{F}\bigl(x,u( \tau,\tau-t, \theta_{-\tau}\omega,u_{0})\bigr) \\& \quad \leq e^{-\sigma t}\bigl( \Vert v_{0} \Vert ^{2}+\bigl(\lambda+\delta^{2}-\beta_{2} \delta\bigr) \Vert u_{0} \Vert ^{2}+ \Vert \Delta u_{0} \Vert ^{2}+2\widetilde{F}(x,u_{0}) \bigr) \\& \qquad {}+ \frac{3}{ (\beta_{1}-\delta)} \int^{\tau}_{\tau-t}e^{\sigma(s- \tau)} \bigl\Vert g(x,s) \bigr\Vert ^{2}\,ds \\& \qquad {}+c \int^{\tau}_{\tau-t}e^{\sigma(s-\tau)}\bigl(1+ \bigl\vert \theta_{-\tau}\omega(s) \bigr\vert ^{2}+ \bigl\vert \theta_{-\tau}\omega(s) \bigr\vert ^{p+1}\bigr)\,ds. \end{aligned}$$
(4.17)
Again, by (3.9), we get
$$ \widetilde{F}(x,u_{0})\leq c\bigl(1+ \Vert u_{0} \Vert ^{2}+ \Vert u_{0} \Vert ^{p+1} \bigr). $$
Thus, for the first term on the right-hand side of (4.17), we have
$$\begin{aligned}& e^{-\sigma t}\bigl( \Vert v_{0} \Vert ^{2}+ \bigl(\lambda+\delta^{2}-\beta_{2}\delta\bigr) \Vert u_{0} \Vert ^{2}+ \Vert \Delta u_{0} \Vert ^{2}+2\widetilde{F}(x,u_{0})\bigr) \\& \quad \leq ce^{- \sigma t}\bigl(1+ \Vert v_{0} \Vert ^{2}+ \Vert u_{0} \Vert ^{2}_{H^{2}}+ \Vert u_{0} \Vert ^{p+1}_{H^{2}}\bigr). \end{aligned}$$
Since \((u_{0},v_{0})^{\top}\in D(\tau-t,\theta_{-t}\omega)\) and \(D\in\mathcal{D}\), then we find
$$ \lim_{t\rightarrow+\infty}e^{-\sigma t}\bigl( \Vert v_{0} \Vert ^{2}+ \Vert u_{0} \Vert ^{2}_{H^{2}}+ \Vert u_{0} \Vert ^{p+1}_{H^{2}}\bigr)=0. $$
Therefore, there exists \(T=T(\tau,\omega,D)>0\) such that, for all \(t\geq T\),
$$ e^{-\sigma t}\bigl(1+ \Vert v_{0} \Vert ^{2}+ \Vert u_{0} \Vert ^{2}_{H^{2}}+ \Vert u_{0} \Vert ^{p+1}_{H^{2}}\bigr) \leq1. $$
(4.18)
For the last term on the right-hand side of (4.17), we find
$$\begin{aligned}& c \int^{\tau}_{\tau-t}e^{\sigma(s-\tau)}\bigl(1+ \bigl\vert \theta_{-\tau} \omega(s) \bigr\vert ^{2}+ \bigl\vert \theta_{-\tau}\omega(s) \bigr\vert ^{p+1}\bigr)\,ds \\& \quad \leq c \int^{0}_{ -t}e^{\sigma s }\bigl(1+ \bigl\vert \omega(s) \bigr\vert ^{2}+ \bigl\vert \omega(s) \bigr\vert ^{p+1}\bigr)\,ds \\& \quad \leq c \int^{0}_{ -\infty}e^{\sigma s }\bigl(1+ \bigl\vert \omega(s) \bigr\vert ^{2}+ \bigl\vert \omega(s) \bigr\vert ^{p+1}\bigr)\,ds \\& \quad \leq \frac{c}{\sigma}+c \int^{0}_{ -\infty}e^{\sigma s }\bigl( \bigl\vert \omega(s) \bigr\vert ^{2}+ \bigl\vert \omega(s) \bigr\vert ^{p+1}\bigr)\,ds. \end{aligned}$$
(4.19)
Notice that \(\omega(s)\) has at most linear growth at \(|s|\rightarrow\infty\), which combines (3.19), we can have
$$ c \int^{\tau}_{\tau-t}e^{\sigma(s-\tau)}\bigl(1+ \bigl\vert \theta_{-\tau}\omega (s) \bigr\vert ^{2}+ \bigl\vert \theta_{-\tau}\omega(s) \bigr\vert ^{p+1}\bigr)\,ds \rightarrow\frac{c}{\sigma}\quad (t\rightarrow\infty). $$
(4.20)
Finally, we estimate the fourth term on the left-hand side of (4.17). Thanks to (3.6), we obtain that, for all \(t\geq0\),
$$ -2\widetilde{F}\bigl(x,u(\tau,\tau-t,\theta_{-\tau} \omega,u_{0})\bigr) \leq2 \int_{\mathbb{R}^{n}}\eta_{3}\,dx. $$
(4.21)
It follows from (4.18)–(4.21) that
$$\begin{aligned}& \bigl\Vert v(\tau,\tau-t,\theta_{-\tau}\omega,v_{0}) \bigr\Vert ^{2} +\bigl(\lambda+ \delta^{2}- \beta_{2}\delta\bigr) \bigl\Vert u(\tau,\tau-t,\theta_{-\tau} \omega,u_{0}) \bigr\Vert ^{2} + \bigl\Vert \Delta u( \tau,\tau-t,\omega,u_{0}) \bigr\Vert ^{2} \\& \quad \leq c+c \int^{\tau}_{-\infty}e^{\sigma(s-\tau)} \bigl\Vert g(x, s) \bigr\Vert ^{2}\,ds+c \int^{0}_{ -\infty}e^{\sigma s }\bigl( \bigl\vert \omega(s) \bigr\vert ^{2}+ \bigl\vert \omega(s) \bigr\vert ^{p+1}\bigr)\,ds. \end{aligned}$$
(4.22)
Thus the proof is completed. □
Lemma 4.2
Under (3.4)–(3.9) and (3.14)–(3.17), for every\(\tau\in\mathbb{R}\), \(\omega\in\varOmega\), \(D=\{D(\tau,\omega):\tau\in\mathbb{R},\omega\in\varOmega\}\in \mathcal{D}\), there exists\(T=T(\tau,\omega,D)>0\)such that, for all\(t\geq T\), the solution of problem (3.10)–(3.11) satisfies
$$ \bigl\Vert A^{\frac{1}{4}}Y\bigl(\tau,\tau-t,\theta_{-\tau} \omega,D(\tau-t, \theta_{-t}\omega)\bigr) \bigr\Vert ^{2}_{E}\leq R_{2}(\tau,\omega), $$
and\(R_{2}(\tau,\omega)\)is given by
$$\begin{aligned} R_{2}(\tau,\omega) =& ce^{-\sigma t}\bigl( \bigl\Vert A^{\frac{1}{4}}v_{0} \bigr\Vert ^{2}+ \bigl\Vert A^{ \frac{1}{4}}u_{0} \bigr\Vert ^{2}+ \bigl\Vert A^{\frac{3}{4}} u_{0} \bigr\Vert ^{2} \bigr) \\ &{}+ c \int^{\tau}_{-\infty}e^{\sigma(s-\tau)} \bigl\Vert g(x,s) \bigr\Vert _{1}^{2}\,ds+c \int^{0}_{-\infty}e^{\sigma s }\bigl(1+ \bigl\vert \omega(s) \bigr\vert ^{2} \bigr)\,ds. \end{aligned}$$
(4.23)
Proof
Taking the inner product of the second equation of (3.10) with \(A^{\frac{1}{2}}v\) in \(L^{2}(\mathbb{R}^{n})\), we find that
$$\begin{aligned}& \frac{1}{2}\frac{d}{dt} \bigl\Vert A^{\frac{1}{4}}v \bigr\Vert ^{2}- \delta \bigl\Vert A^{ \frac{1}{4}}v \bigr\Vert ^{2}+\bigl(\lambda+\delta^{2}\bigr) \bigl(u,A^{\frac{1}{2}}v\bigr)+ \bigl(Au,A^{ \frac{1}{2}}v\bigr)+ \bigl(f(x,u),A^{\frac{1}{2}}v\bigr) \\& \quad =\bigl(g(x,t),A^{\frac{1}{2}}v\bigr)-\bigl(h\bigl(v+ \phi\omega(t)- \delta u\bigr),A^{ \frac{1}{2}}v\bigr)+\delta\bigl(\phi,A^{\frac{1}{2}}v \bigr)\omega(t). \end{aligned}$$
(4.24)
Similar to the proof of Lemma 4.1, we have the following estimates:
$$\begin{aligned}& -\bigl(h\bigl(v+ \phi\omega(t)-\delta u\bigr),A^{\frac{1}{2}}v\bigr) \\& \quad = -\bigl(h\bigl(v+ \phi\omega(t)-\delta u\bigr)-h(0),A^{\frac{1}{2}}v \bigr) \\& \quad = -\bigl(h'(\vartheta) \bigl(v+ \phi\omega(t)-\delta u \bigr),A^{\frac{1}{2}}v\bigr) \\& \quad \leq -\beta_{1} \bigl\Vert A^{\frac{1}{4}}v \bigr\Vert ^{2}-\bigl(h'(\vartheta) \bigl( \phi \omega(t)-\delta u\bigr),A^{\frac{1}{2}}v\bigr) \\& \quad \leq -\beta_{1} \bigl\Vert A^{\frac{1}{4}}v \bigr\Vert ^{2}+\beta_{2} \bigl\vert \omega(t) \bigr\vert \bigl\Vert A^{ \frac{1}{4}}\phi \bigr\Vert \bigl\Vert A^{\frac{1}{4}}v \bigr\Vert +h'(\vartheta)\delta\bigl(u,A^{ \frac{1}{2}}v\bigr) \\& \quad \leq -\beta_{1} \bigl\Vert A^{\frac{1}{4}}v \bigr\Vert ^{2}+ \frac{\beta_{1}-\delta}{6} \bigl\Vert A^{\frac{1}{4}}v \bigr\Vert ^{2}+c \bigl\vert \omega(t) \bigr\vert ^{2} \bigl\Vert A^{\frac{1}{4}}\phi \bigr\Vert ^{2}+h'( \vartheta)\delta\bigl(u,A^{\frac{1}{2}}v\bigr), \end{aligned}$$
(4.25)
$$\begin{aligned}& h'(\vartheta)\delta\bigl(u,A^{\frac{1}{2}}v\bigr) \\& \quad = h'(\vartheta)\delta\bigl(u,A^{\frac{1}{2}}u_{t}- \omega(t)A^{ \frac{1}{2}}\phi+\delta A^{\frac{1}{2}}u\bigr) \\& \quad \leq \beta_{2}\delta\cdot\frac{1}{2} \frac{d}{dt} \bigl\Vert A^{\frac{1}{4}}u \bigr\Vert ^{2}+\beta_{2}\delta^{2} \bigl\Vert A^{\frac{1}{4}}u \bigr\Vert ^{2}+ \beta_{2} \delta \bigl\vert \omega(t) \bigr\vert \bigl\Vert A^{\frac{1}{4}}\phi \bigr\Vert \bigl\Vert A^{\frac{1}{4}}u \bigr\Vert \\& \quad \leq \beta_{2}\delta\cdot\frac{1}{2} \frac{d}{dt} \bigl\Vert A^{\frac{1}{4}}u \bigr\Vert ^{2}+\beta_{2}\delta^{2} \bigl\Vert A^{\frac{1}{4}}u \bigr\Vert ^{2} \\& \qquad {}+\frac{1}{6} \delta \bigl(\lambda+\delta^{2}-\beta_{2}\delta\bigr) \bigl\Vert A^{\frac{1}{4}}u \bigr\Vert ^{2}+c \bigl\vert \omega(t) \bigr\vert ^{2} \bigl\Vert A^{\frac{1}{4}}\phi \bigr\Vert ^{2}, \end{aligned}$$
(4.26)
$$\begin{aligned}& \bigl(\lambda+\delta^{2}\bigr) \bigl(u,A^{\frac{1}{2}}v\bigr) \\& \quad = \bigl(\lambda+\delta^{2}\bigr) \bigl(u,A^{\frac{1}{2}}u_{t}- \omega(t)A^{ \frac{1}{2}}\phi+\delta A^{\frac{1}{2}}u\bigr) \\& \quad \geq \frac{1}{2}\bigl(\lambda+\delta^{2}\bigr) \frac{d}{dt} \bigl\Vert A^{\frac{1}{4}}u \bigr\Vert ^{2}+\delta\bigl(\lambda+\delta^{2}\bigr) \bigl\Vert A^{\frac{1}{4}}u \bigr\Vert ^{2}-\bigl( \lambda+ \delta^{2}\bigr) \bigl\vert \omega(t) \bigr\vert \bigl\Vert A^{\frac{1}{4}}\phi \bigr\Vert \bigl\Vert A^{ \frac{1}{4}}u \bigr\Vert \\& \quad \geq \frac{1}{2}\bigl(\lambda+\delta^{2}\bigr) \frac{d}{dt} \bigl\Vert A^{\frac{1}{4}}u \bigr\Vert ^{2}+\delta\bigl(\lambda+\delta^{2}\bigr) \bigl\Vert A^{\frac{1}{4}}u \bigr\Vert ^{2} \\& \qquad {}- \frac{1}{6}\delta \bigl(\lambda+\delta^{2}-\beta_{2}\delta\bigr) \bigl\Vert A^{ \frac{1}{4}}u \bigr\Vert ^{2}-c \bigl\vert \omega(t) \bigr\vert ^{2} \bigl\Vert A^{\frac{1}{4}}\phi \bigr\Vert ^{2}, \end{aligned}$$
(4.27)
$$\begin{aligned}& \begin{aligned}[b] \bigl(Au,A^{\frac{1}{2}}v\bigr) &=\bigl(A u,A^{\frac{1}{2}} u_{t}-\omega(t)A^{ \frac{1}{2}}\phi+\delta A^{\frac{1}{2}} u \bigr) \\ &\geq \frac{1}{2}\frac{d}{dt} \bigl\Vert A^{\frac{3}{4}} u \bigr\Vert ^{2}+\delta \bigl\Vert A^{ \frac{3}{4}} u \bigr\Vert ^{2}- \bigl\vert \omega(t) \bigr\vert \bigl\Vert A^{\frac{3}{4}} \phi \bigr\Vert \bigl\Vert A^{ \frac{3}{4}} u \bigr\Vert \\ &\geq \frac{1}{2}\frac{d}{dt} \bigl\Vert A^{\frac{3}{4}} u \bigr\Vert ^{2}+ \frac{\delta}{2} \bigl\Vert A^{\frac{3}{4}} u \bigr\Vert ^{2}-\frac{1}{2\delta} \bigl\vert \omega(t) \bigr\vert ^{2} \bigl\Vert A^{\frac{3}{4}}\phi \bigr\Vert ^{2}, \end{aligned} \end{aligned}$$
(4.28)
$$\begin{aligned}& \delta\bigl(\phi\omega(t),A^{\frac{1}{2}}v\bigr)\leq\delta \bigl\vert \omega(t) \bigr\vert \bigl\Vert A^{ \frac{1}{4}}\phi \bigr\Vert \bigl\Vert A^{\frac{1}{4}}v \bigr\Vert \leq c \bigl\Vert A^{\frac{1}{4}} \phi \bigr\Vert ^{2} \bigl\vert \omega(t) \bigr\vert ^{2}+\frac{\beta_{1}-\delta}{6} \bigl\Vert A^{\frac{1}{4}}v \bigr\Vert ^{2}, \end{aligned}$$
(4.29)
$$\begin{aligned}& \bigl(g,A^{\frac{1}{2}}v\bigr)\leq \Vert g \Vert _{1} \bigl\Vert A^{\frac{1}{4}}v \bigr\Vert \leq \frac{3}{2(\beta_{1}-\delta)} \Vert g \Vert ^{2}_{1}+ \frac{\beta_{1}-\delta}{6} \bigl\Vert A^{\frac{1}{4}}v \bigr\Vert ^{2}. \end{aligned}$$
(4.30)
For the last term on the left-hand side of (4.24), thanks to (3.7), we have
$$\begin{aligned} - \bigl(f(x,u),A^{\frac{1}{2}}v \bigr) \leq& \bigl\vert \bigl(f(x,u),A^{\frac{1}{2}}v \bigr) \bigr\vert \\ =& \biggl\vert \int_{\mathbb{R}^{n}} \frac{\partial}{\partial x}f(x,u)\cdot A^{ \frac{1}{4}}v \,dx+ \int_{\mathbb{R}^{n}} \frac{\partial}{\partial u}f(x,u) \cdot A^{\frac{1}{4}}u\cdot A^{\frac{1}{4}}v \,dx \biggr\vert \\ \leq& \int_{\mathbb{R}^{n}} \biggl\vert \frac{\partial}{\partial x}f(x,u) \biggr\vert \cdot \bigl\vert A^{\frac{1}{4}}v \bigr\vert \,dx+ \int_{\mathbb{R}^{n}} \biggl\vert \frac{\partial}{\partial u}f(x,u) \biggr\vert \cdot \bigl\vert A^{\frac{1}{4}}u \bigr\vert \cdot \bigl\vert A^{ \frac{1}{4}}v \bigr\vert \,dx \\ \leq& \int_{\mathbb{R}^{n}} \vert \eta_{4} \vert \cdot \bigl\vert A^{\frac{1}{4}}v \bigr\vert \,dx+ \beta \int_{\mathbb{R}^{n}} \bigl\vert A^{\frac{1}{4}}u \bigr\vert \cdot \bigl\vert A^{\frac{1}{4}}v \bigr\vert \,dx \\ \leq& \Vert \eta_{4} \Vert \bigl\Vert A^{\frac{1}{4}}v \bigr\Vert +\beta \bigl\Vert A^{\frac{1}{4}}u \bigr\Vert \bigl\Vert A^{ \frac{1}{4}}v \bigr\Vert \\ \leq& c + \biggl(\delta+ \frac{3\beta^{2}}{2\delta(\lambda+\delta^{2}-\beta_{2}\delta)} \biggr) \bigl\Vert A^{\frac{1}{4}}v \bigr\Vert ^{2} \\ &{}+\frac{1}{6}\delta \bigl(\lambda+\delta^{2}- \beta_{2}\delta\bigr) \bigl\Vert A^{\frac{1}{4}}u \bigr\Vert ^{2}. \end{aligned}$$
(4.31)
Plugging (4.25)–(4.31) into (4.24) and together with (3.15), we obtain
$$\begin{aligned}& \frac{1}{2}\frac{d}{dt}\bigl( \bigl\Vert A^{\frac{1}{4}}v \bigr\Vert ^{2}+\bigl(\lambda+ \delta^{2}- \beta_{2}\delta\bigr) \bigl\Vert A^{\frac{1}{4}}u \bigr\Vert ^{2}+ \bigl\Vert A^{\frac{3}{4}} u \bigr\Vert ^{2} \bigr) \\& \qquad {} +\delta\bigl( \bigl\Vert A^{\frac{1}{4}}v \bigr\Vert ^{2}+\bigl(\lambda+\delta^{2}-\beta_{2} \delta\bigr) \bigl\Vert A^{\frac{1}{4}}u \bigr\Vert ^{2}+ \bigl\Vert A^{\frac{3}{4}}u \bigr\Vert ^{2}\bigr) \\& \quad \leq \frac{\delta}{2} \bigl\Vert A^{\frac{1}{4}}v \bigr\Vert ^{2}+\frac{\delta}{2}\bigl( \lambda+\delta^{2}- \beta_{2}\delta\bigr) \bigl\Vert A^{\frac{1}{4}}u \bigr\Vert ^{2} + \frac{\delta}{2} \bigl\Vert A^{\frac{3}{4}} u \bigr\Vert ^{2} +c\bigl(1+ \bigl\vert \omega(t) \bigr\vert ^{2}+ \Vert g \Vert ^{2}_{1}\bigr), \end{aligned}$$
(4.32)
then
$$\begin{aligned}& \frac{d}{dt}\bigl( \bigl\Vert A^{\frac{1}{4}}v \bigr\Vert ^{2}+\bigl(\lambda+\delta^{2}-\beta_{2} \delta\bigr) \bigl\Vert A^{\frac{1}{4}}u \bigr\Vert ^{2}+ \bigl\Vert A^{\frac{3}{4}} u \bigr\Vert ^{2} \bigr) \\& \qquad {} +\sigma\bigl( \bigl\Vert A^{\frac{1}{4}}v \bigr\Vert ^{2}+\bigl(\lambda+\delta^{2}-\beta_{2} \delta\bigr) \bigl\Vert A^{\frac{1}{4}}u \bigr\Vert ^{2}+ \bigl\Vert A^{\frac{3}{4}} u \bigr\Vert ^{2}\bigr) \\& \quad \leq c\bigl(1+ \bigl\vert \omega(t) \bigr\vert ^{2}+ \Vert g \Vert ^{2}_{1} \bigr). \end{aligned}$$
(4.33)
Multiplying (4.33) by \(e^{\sigma t}\) and then integrating over \((\tau-t,\tau)\), we have
$$\begin{aligned}& e^{\sigma\tau} \bigl( \bigl\Vert A^{\frac{1}{4}}v(\tau,\tau-t, \omega,v_{0}) \bigr\Vert ^{2}+\bigl( \lambda+ \delta^{2}-\beta_{2}\delta\bigr) \bigl\Vert A^{\frac{1}{4}} u(\tau, \tau-t,\omega,u_{0}) \bigr\Vert ^{2} \\& \qquad {}+ \bigl\Vert A^{\frac{3}{4}} u(\tau,\tau-t, \omega,u_{0}) \bigr\Vert ^{2} \bigr) \\& \quad \leq e^{\sigma(\tau-t)}\bigl( \bigl\Vert A^{\frac{1}{4}}v_{0} \bigr\Vert ^{2}+\bigl(\lambda+ \delta^{2}- \beta_{2}\delta\bigr) \bigl\Vert A^{\frac{1}{4}} u_{0} \bigr\Vert ^{2}+ \bigl\Vert A^{ \frac{3}{4}} u_{0} \bigr\Vert ^{2} \bigr) \\& \qquad {}+c \int^{\tau}_{\tau-t}e^{\sigma s}\bigl(1+ \bigl\vert \omega(s) \bigr\vert ^{2}+ \bigl\Vert g(x,s) \bigr\Vert _{1}^{2} \bigr)\,ds. \end{aligned}$$
Replacing ω by \(\theta_{-\tau}\omega\) in the above, we obtain, for every \(t\in\mathbb{R}^{+}\), \(\tau\in\mathbb{R}\), and \(\omega\in \varOmega\),
$$\begin{aligned}& \bigl\Vert A^{\frac{1}{4}}v(\tau,\tau-t,\theta_{-\tau} \omega,v_{0}) \bigr\Vert ^{2}+\bigl( \lambda+ \delta^{2}-\beta_{2}\delta\bigr) \bigl\Vert A^{\frac{1}{4}}u(\tau,\tau-t, \theta_{-\tau}\omega,u_{0}) \bigr\Vert ^{2} \\& \qquad {} + \bigl\Vert A^{\frac{3}{4}} u(\tau,\tau-t,\theta_{-\tau} \omega,u_{0}) \bigr\Vert ^{2} \\& \quad \leq e^{-\sigma t}\bigl( \bigl\Vert A^{\frac{1}{4}}v_{0} \bigr\Vert ^{2}+\bigl(\lambda+\delta^{2}- \beta_{2}\delta\bigr) \bigl\Vert A^{\frac{1}{4}}u_{0} \bigr\Vert ^{2}+ \bigl\Vert A^{\frac{3}{4}} u_{0} \bigr\Vert ^{2} \bigr) \\& \qquad {}+ \frac{3}{ (\beta_{1}-\delta)} \int^{\tau}_{\tau-t}e^{\sigma(s- \tau)} \bigl\Vert g(x,s) \bigr\Vert _{1}^{2}\,ds+c \int^{\tau}_{\tau-t}e^{\sigma(s-\tau)}\bigl(1+ \bigl\vert \theta_{-\tau}\omega(s) \bigr\vert ^{2} \bigr)\,ds \\& \quad \leq ce^{-\sigma t}\bigl( \bigl\Vert A^{\frac{1}{4}}v_{0} \bigr\Vert ^{2}+ \bigl\Vert A^{\frac{1}{4}}u_{0} \bigr\Vert ^{2}+ \bigl\Vert A^{\frac{3}{4}} u_{0} \bigr\Vert ^{2} \bigr) \\& \qquad {}+ c \int^{\tau}_{-\infty}e^{\sigma(s-\tau)} \bigl\Vert g(x,s) \bigr\Vert _{1}^{2}\,ds+c \int^{0}_{-\infty}e^{\sigma s }\bigl(1+ \bigl\vert \omega(s) \bigr\vert ^{2} \bigr)\,ds. \end{aligned}$$
(4.34)
Thus the proof is completed. □
Lemma 4.3
Under (3.4)–(3.9) and (3.14)–(3.17), for every\(\eta>0\), \(\tau\in\mathbb{R}\), \(\omega\in\varOmega\), \(D=\{D(\tau,\omega):\tau\in\mathbb{R},\omega\in\varOmega\}\in \mathcal{D}\), there exist\(T=T(\tau,\omega,D,\eta)>0\), \(K=K(\tau,\omega,\eta)\geq1\)such that, for all\(t\geq T\), \(k\geq K\), the solution of problem (3.10)–(3.11) satisfies
$$ \bigl\Vert Y\bigl(\tau,\tau-t,\theta_{-\tau}\omega,D(\tau-t, \theta_{-t} \omega)\bigr) \bigr\Vert ^{2}_{E(\mathbb{R}^{n}\setminus\mathbb{B}_{k})} \leq\eta, $$
(4.35)
where for\(k\geq1\), \(\mathbb{B}_{k}=\{x\in\mathbb{R}^{n}:|x| \leq k\}\)and\(\mathbb{R}^{n}\setminus\mathbb{B}_{k}\)is the complement of\(\mathbb{B}_{k}\).
Proof
Choose a smooth function ρ such that \(0\leq\rho\leq1\) for \(s\in\mathbb{R}\), and
$$ \rho(s)= \textstyle\begin{cases} 0, & \text{if } 0\leq \vert s \vert \leq1, \\ 1, & \text{if } \vert s \vert \geq2, \end{cases} $$
(4.36)
and there exist constants \(\mu_{1}\), \(\mu_{2}\), \(\mu_{3}\), \(\mu _{4}\) such that \(|\rho'(s)|\leq\mu_{1}\), \(|\rho''(s)|\leq\mu_{2}\), \(|\rho'''(s)| \leq\mu_{3}\), \(|\rho''''(s)|\leq\mu_{4}\) for \(s\in\mathbb{R}\). Taking the inner product of the second equation of (3.10) with \(\rho(\frac{|x|^{2}}{k^{2}})v\) in \(L^{2}(\mathbb{R}^{n})\), we obtain
$$\begin{aligned}& \frac{1}{2}\frac{d}{dt} \int_{\mathbb{R}^{n}}\rho\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert v \vert ^{2}\,dx- \delta \int_{\mathbb{R}^{n}}\rho\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert v \vert ^{2}\,dx \\& \qquad {} +\bigl(\lambda+\delta^{2}\bigr) \int_{\mathbb{R}^{n}}\rho\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)uv\,dx+ \int_{\mathbb{R}^{n}}(Au)\rho\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)v\,dx+ \int_{\mathbb{R}^{n}}\rho\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)f(x,u)v\,dx \\& \quad =\delta \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\phi \omega(t) v\,dx - \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) h\bigl(v+ \phi\omega(t)-\delta u\bigr)v\,dx \\& \qquad {}+ \int_{\mathbb{R}^{n}}\rho\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)g(x,t)v\,dx. \end{aligned}$$
(4.37)
Similar to (4.5), we have
$$\begin{aligned}& - \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) h\bigl(v+ \phi\omega(t)- \delta u\bigr)v\,dx \\& \quad =- \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \bigl(h \bigl(v+\phi \omega(t)-\delta u\bigr)-h(0)\bigr)v\,dx \\& \quad \leq-\beta_{1} \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert v \vert ^{2}\,dx+h'( \vartheta)\delta \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)uv\,dx \\& \qquad {}+ \beta_{2} \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert \phi \vert \bigl\vert \omega(t) \bigr\vert \vert v \vert \,dx. \end{aligned}$$
(4.38)
Taking (4.38) into (4.37), we have
$$\begin{aligned}& \frac{1}{2}\frac{d}{dt} \int_{\mathbb{R}^{n}}\rho\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert v \vert ^{2}\,dx-(\delta-\beta_{1}) \int_{\mathbb{R}^{n}} \rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert v \vert ^{2}\,dx+ \int_{\mathbb{R}^{n}}(Au) \rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)v\,dx \\& \qquad {} +\bigl(\lambda+\delta^{2}-h'(\vartheta) \delta\bigr) \int_{\mathbb{R}^{n}} \rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)uv\,dx + \int_{\mathbb{R}^{n}}\rho\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)f(x,u)v\,dx \\& \quad \leq (1+\delta+\beta_{2}) \int_{\mathbb{R}^{n}}\rho\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert \phi \vert \bigl\vert \omega(t) \bigr\vert \vert v \vert \,dx+ \int_{\mathbb{R}^{n}} \rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)g(x,t)v\,dx \\& \quad \leq \frac{\beta_{1}-\delta}{3} \int_{\mathbb{R}^{n}}\rho\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert v \vert ^{2}\,dx+c \int_{\mathbb{R}^{n}}\rho\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert \phi \vert ^{2} \bigl\vert \omega(t) \bigr\vert ^{2} \,dx \\& \qquad {}+ \int_{\mathbb{R}^{n}} \rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)g(x,t)v\,dx. \end{aligned}$$
(4.39)
For the fourth term on the left-hand side of (4.39), we have
$$\begin{aligned}& \bigl(\lambda+\delta^{2}-h'(\vartheta)\delta\bigr) \int_{\mathbb{R}^{n}} \rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)uv\,dx \\& \quad = \bigl(\lambda+\delta^{2}-h'(\vartheta)\delta \bigr) \int_{\mathbb{R}^{n}} \rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)u\biggl( \frac{du}{dt}+\delta u-\phi\omega(t)\biggr)\,dx \\& \quad = \bigl(\lambda+\delta^{2}-h'(\vartheta)\delta \bigr) \int_{\mathbb{R}^{n}} \rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \biggl( \frac{1}{2}\frac{d}{dt} u^{2}+ \delta u^{2}- \phi\omega(t)u\biggr)\,dx \\& \quad \geq \bigl(\lambda+\delta^{2}-\beta_{2}\delta \bigr) \biggl(\frac{1}{2} \frac{d}{dt} \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert u \vert ^{2} \,dx+ \delta \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert u \vert ^{2}\,dx\biggr) \\& \qquad {} -\bigl(\lambda+\delta^{2}-\beta_{1}\delta \bigr) \int_{\mathbb{R}^{n}}\rho\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert \phi \vert \bigl\vert \omega(t) \bigr\vert \vert u \vert \,dx \\& \quad \geq \bigl(\lambda+\delta^{2}-\beta_{2}\delta \bigr) \biggl(\frac{1}{2} \frac{d}{dt} \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert u \vert ^{2} \,dx+ \delta \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert u \vert ^{2}\,dx\biggr) \\& \qquad {} -\frac{\delta}{2}\bigl(\lambda+\delta^{2}- \beta_{2}\delta\bigr) \int_{ \mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert u \vert ^{2}\,dx-c \int_{ \mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert \phi \vert ^{2} \bigl\vert \omega(t) \bigr\vert ^{2} \,dx. \end{aligned}$$
(4.40)
For the third term on the left-hand side of (4.39), we have
$$\begin{aligned}& \int_{\mathbb{R}^{n}}(Au)\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)v\,dx \\& \quad = \int_{\mathbb{R}^{n}}(Au)\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \biggl( \frac{du}{dt}+\delta u-\phi\omega(t)\biggr)\,dx \\& \quad = \int_{\mathbb{R}^{n}}\bigl(\Delta^{2}u\bigr)\rho\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \biggl( \frac{du}{dt}+\delta u-\phi\omega(t) \biggr)\,dx \\& \quad = \int_{\mathbb{R}^{n}}(\Delta u)\Delta\biggl(\rho\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \biggl( \frac{du}{dt}+\delta u-\phi\omega(t) \biggr)\biggr)\,dx \\& \quad = \int_{\mathbb{R}^{n}}(\Delta u) \biggl(\biggl(\frac{2}{k^{2}} \rho'\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)+\frac{4x^{2}}{k^{4}} \rho''\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\biggr) \biggl(\frac{du}{dt}+\delta u-\phi\omega(t)\biggr) \\& \qquad {} +2\cdot\frac{2 \vert x \vert }{k^{2}}\rho'\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\nabla\biggl( \frac{du}{dt}+\delta u-\phi \omega(t)\biggr)+\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \Delta\biggl( \frac{du}{dt}+\delta u-\phi\omega(t)\biggr)\biggr)\,dx \\& \quad \geq - \int_{k< x< \sqrt{2}k}\biggl(\frac{2\mu_{1}}{k^{2}}+ \frac{4\mu_{2}x^{2}}{k^{4}} \biggr) \bigl\vert (\Delta u)v \bigr\vert \,dx- \int_{k< x< \sqrt{2}k} \frac{4\mu_{1}x}{k^{2}} \bigl\vert (\Delta u) ( \nabla v) \bigr\vert \,dx \\& \qquad {} +\frac{1}{2}\frac{d}{dt} \int_{\mathbb{R}^{n}}\rho\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert \Delta u \vert ^{2}\,dx+\delta \int_{\mathbb{R}^{n}} \rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert \Delta u \vert ^{2}\,dx \\& \qquad {}- \int_{\mathbb{R}^{n}} \rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert \Delta u \vert \vert \Delta\phi \vert \bigl\vert \omega(t) \bigr\vert \,dx \\& \quad \geq - \int_{\mathbb{R}^{n}}\biggl(\frac{2\mu_{1}+8\mu_{2}}{k^{2}}\biggr) \bigl\vert ( \Delta u)v \bigr\vert \,dx- \int_{\mathbb{R}^{n}}\frac{4\sqrt{2}\mu_{1}}{k} \bigl\vert ( \Delta u) ( \nabla v) \bigr\vert \,dx \\& \qquad {}+\frac{1}{2}\frac{d}{dt} \int_{\mathbb{R}^{n}} \rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert \Delta u \vert ^{2}\,dx +\delta \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert \Delta u \vert ^{2}\,dx \\& \qquad {}- \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert \Delta u \vert \vert \Delta \phi \vert \bigl\vert \omega(t) \bigr\vert \,dx \\& \quad \geq -\frac{\mu_{1}+4\mu_{2}}{k^{2}}\bigl( \Vert \Delta u \Vert ^{2}+ \Vert v \Vert ^{2}\bigr)- \frac{4\sqrt{2}\mu_{1}}{k} \Vert \Delta u \Vert \Vert \nabla v \Vert +\frac{1}{2} \frac{d}{dt} \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert \Delta u \vert ^{2}\,dx \\& \qquad {} +\delta \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert \Delta u \vert ^{2}\,dx- \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert \Delta u \vert \vert \Delta \phi \vert \bigl\vert \omega(t) \bigr\vert \,dx \\& \quad \geq -\frac{\mu_{1}+4\mu_{2}}{k^{2}}\bigl( \Vert \Delta u \Vert ^{2}+ \Vert v \Vert ^{2}\bigr)- \frac{2\sqrt{2}\mu_{1}}{k} \bigl( \Vert \Delta u \Vert ^{2}+ \Vert \nabla v \Vert ^{2}\bigr) \\& \qquad {} + \frac{1}{2}\frac{d}{dt} \int_{\mathbb{R}^{n}}\rho\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert \Delta u \vert ^{2}\,dx +\delta \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert \Delta u \vert ^{2}\,dx \\& \qquad {}- \frac{\delta}{2} \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert \Delta u \vert ^{2}\,dx-c \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert \Delta\phi \vert ^{2} \bigl\vert \omega(t) \bigr\vert ^{2}\,dx, \end{aligned}$$
(4.41)
$$\begin{aligned}& \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)f(x,u)v\,dx \\& \quad = \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)f(x,u) \biggl( \frac{du}{dt}+\delta u-\phi\omega(t)\biggr)\,dx \\& \quad = \frac{d}{dt} \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)F(x,u)\,dx+ \delta \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)f(x,u)u\,dx \\& \qquad {}- \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)f(x,u) \phi \omega(t) \,dx. \end{aligned}$$
(4.42)
Similar to (4.12) and (4.13) in Lemma 4.1, we have
$$\begin{aligned}& \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)f(x,u)u\,dx \geq c_{2} \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)F(x,u)\,dx + \int_{ \mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \eta_{2}\,dx, \end{aligned}$$
(4.43)
$$\begin{aligned}& \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)f(x,u) \phi \omega(t)\,dx \\& \quad \leq \frac{1}{2} \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert \eta_{1} \vert ^{2}\,dx+ \frac{1}{2} \int_{\mathbb{R}^{n}}\rho\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert \phi \vert ^{2} \bigl\vert \omega(t) \bigr\vert ^{2} \,dx \\& \qquad {}+\frac{\delta c_{2}}{2} \int_{\mathbb{R}^{n}}\rho\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \bigl(F(x,u)+\eta_{3}\bigr)\,dx+c \int_{\mathbb{R}^{n}} \rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert \phi \vert ^{p+1} \bigl\vert \omega(t) \bigr\vert ^{p+1}\,dx, \end{aligned}$$
(4.44)
$$\begin{aligned}& \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)g(x,t)v\,dx \\& \quad \leq \frac{3}{2(\beta_{1}-\delta)} \int_{\mathbb{R}^{n}}\rho\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \bigl\vert g(x,t) \bigr\vert ^{2}\,dx +\frac{\beta_{1}-\delta}{6} \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert v \vert ^{2}\,dx. \end{aligned}$$
(4.45)
By (4.38)–(4.45), we have
$$\begin{aligned}& \frac{1}{2}\frac{d}{dt} \int_{\mathbb{R}^{n}}\rho\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \bigl( \vert v \vert ^{2}+\bigl(\lambda+\delta^{2}- \beta_{2} \delta\bigr) \vert u \vert ^{2}+ \vert \Delta u \vert ^{2}+2F(x,u)\bigr)\,dx \\& \qquad {} +\delta \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \bigl( \vert v \vert ^{2}+\bigl( \lambda+\delta^{2}- \beta_{2}\delta\bigr) \vert u \vert ^{2}+ \vert \Delta u \vert ^{2}\bigr)\,dx \\& \qquad {}+ \frac{\delta c_{2}}{2} \int_{\mathbb{R}^{n}}\rho\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)F(x,u)\,dx \\& \quad \leq \frac{\delta}{2} \int_{\mathbb{R}^{n}}\rho\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert v \vert ^{2}\,dx+\frac{2\delta-\beta_{1}}{2} \int_{ \mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert v \vert ^{2}\,dx \\& \qquad {}+ \frac{\delta}{2}\bigl(\lambda+ \delta^{2}-\beta_{2}\delta\bigr) \int_{ \mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert u \vert ^{2}\,dx \\& \qquad {} +\frac{\delta}{2} \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \vert \Delta u \vert ^{2}\,dx+ \frac{\mu_{1}+4\mu_{2}}{k^{2}}\bigl( \Vert \Delta u \Vert ^{2}+ \Vert v \Vert ^{2}\bigr) \\& \qquad {}+ \frac{2\sqrt{2}\mu_{1}}{k}\bigl( \Vert \Delta u \Vert ^{2}+ \Vert \nabla v \Vert ^{2}\bigr) \\& \qquad {} +c \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \bigl( \vert \eta_{1} \vert ^{2}+ \vert \eta_{2} \vert + \vert \eta_{3} \vert + \vert g \vert ^{2}+ \bigl\vert \omega(t) \bigr\vert ^{p+1} \vert \phi \vert ^{p+1}\bigr)\,dx \\& \qquad {} +c \bigl\vert \omega(t) \bigr\vert ^{2} \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \bigl( \vert \phi \vert ^{2}+ \vert \Delta\phi \vert ^{2} \bigr)\,dx. \end{aligned}$$
(4.46)
Since \(\phi\in H^{2}(\mathbb{R}^{n})\), \(\eta_{1} \in L^{2}( \mathbb{R}^{n})\), \(\eta_{2} \in L^{1}(\mathbb{R}^{n})\), \(\eta_{3} \in L^{1}(\mathbb{R}^{n})\), we obtain that there exists \(K_{1}=K_{1}(\tau,\eta)\geq1\) such that, for all \(k\geq K_{1}\),
$$\begin{aligned}& c\int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \bigl( \vert \eta_{1} \vert ^{2}+ \vert \eta_{2} \vert + \vert \eta_{3} \vert + \bigl\vert \omega(t) \bigr\vert ^{p+1} \vert \phi \vert ^{p+1} \bigr)\,dx \\& \qquad {}+c \bigl\vert \omega(t) \bigr\vert ^{2} \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \bigl( \vert \phi \vert ^{2}+ \vert \Delta\phi \vert ^{2} \bigr)\,dx \\& \quad = c \int_{ \vert x \vert \geq k}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \bigl( \vert \eta_{1} \vert ^{2}+ \vert \eta_{2} \vert + \vert \eta_{3} \vert + \bigl\vert \omega(t) \bigr\vert ^{p+1} \vert \phi \vert ^{p+1} \bigr)\,dx \\& \qquad {}+c \bigl\vert \omega(t) \bigr\vert ^{2} \int_{ \vert x \vert \geq k}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \bigl( \vert \phi \vert ^{2}+ \vert \Delta \phi \vert ^{2}\bigr)\,dx \\& \quad \leq c \int_{ \vert x \vert \geq k} \bigl( \vert \eta_{1} \vert ^{2}+ \vert \eta_{2} \vert + \vert \eta_{3} \vert + \bigl\vert \omega(t) \bigr\vert ^{p+1} \vert \phi \vert ^{p+1}\bigr)\,dx \\& \qquad {}+c \bigl\vert \omega(t) \bigr\vert ^{2} \int_{ \vert x \vert \geq k} \bigl( \vert \phi \vert ^{2}+ \vert \Delta\phi \vert ^{2}\bigr)\,dx \\& \quad \leq c\eta\bigl(1+ \bigl\vert \omega(t) \bigr\vert ^{2}+ \bigl\vert \omega(t) \bigr\vert ^{p+1}\bigr), \end{aligned}$$
(4.47)
along with
$$ c \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}} \biggr)g^{2}(x,t)\,dx\leq c \int_{ \vert x \vert \geq k}g^{2}(x,t)\,dx, $$
(4.48)
we have that, for all \(k\geq K_{1}\),
$$\begin{aligned}& \frac{d}{dt} \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \bigl( \vert v \vert ^{2}+\bigl( \lambda+\delta^{2}- \beta_{2}\delta\bigr) \vert u \vert ^{2}+ \vert \Delta u \vert ^{2}+2F(x,u)\bigr)\,dx \\& \qquad {} +\sigma \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \bigl( \vert v \vert ^{2}+\bigl( \lambda+\delta^{2}- \beta_{2}\delta\bigr) \vert u \vert ^{2}+ \vert \Delta u \vert ^{2}+2F(x,u)\bigr)\,dx \\& \quad \leq \frac{2\mu_{1}+8\mu_{2}}{k^{2}}\bigl( \Vert \Delta u \Vert ^{2}+ \Vert v \Vert ^{2}\bigr) + \frac{4\sqrt{2}\mu_{1}}{k}\bigl( \Vert \Delta u \Vert ^{2}+ \Vert \nabla v \Vert ^{2}\bigr) \\& \qquad {} +c\eta\bigl(1+ \bigl\vert \omega(t) \bigr\vert ^{2}+ \bigl\vert \omega(t) \bigr\vert ^{p+1}\bigr)+c \int_{ \vert x \vert \geq k}g^{2}(x,t)\,dx. \end{aligned}$$
(4.49)
Multiplying (4.49) by \(e^{\sigma t}\) and then integrating over \((\tau-t,\tau)\), we find
$$\begin{aligned}& \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \bigl( \bigl\vert v(\tau,\tau-t, \omega,v_{0}) \bigr\vert ^{2}+ \bigl(\lambda+\delta^{2}-\beta_{2}\delta\bigr) \bigl\vert u(\tau, \tau-t,\omega,u_{0}) \bigr\vert ^{2} \\& \qquad {} + \bigl\vert \Delta u(\tau,\tau-t,\omega,u_{0}) \bigr\vert ^{2}+2F\bigl(x,u( \tau,\tau-t,\omega,u_{0})\bigr) \bigr)\,dx \\& \quad \leq e^{-\sigma t} \int_{\mathbb{R}^{n}}\rho\biggl(\frac { \vert x \vert ^{2}}{k^{2}}\biggr) \bigl( \vert v_{0} \vert ^{2}+\bigl( \lambda+ \delta^{2}-\beta_{2}\delta\bigr) \vert u_{0} \vert ^{2} + \vert \Delta u_{0} \vert ^{2}+2F(x,u_{0})\bigr)\,dx \\& \qquad {} + \frac{2\mu_{1}+8\mu_{2}}{k^{2}} \int^{\tau}_{\tau-t}e^{\sigma(s- \tau)}\bigl( \bigl\vert \Delta u(s,\tau-t,\omega,u_{0}) \bigr\vert ^{2}+ \bigl\vert v(s,\tau-t,\omega ,v_{0}) \bigr\vert ^{2}\bigr)\,ds \\& \qquad {} +\frac{4\sqrt{2}\mu_{1}}{k} \int^{\tau}_{\tau-t}e^{\sigma(s-\tau)}\bigl( \bigl\vert \Delta u(s,\tau-t,\omega,u_{0}) \bigr\vert ^{2}+ \bigl\vert \nabla v(s,\tau-t,\omega ,v_{0}) \bigr\vert ^{2}\bigr)\,ds \\& \qquad {} +c\frac{\eta}{\sigma}+c\eta \int^{\tau}_{\tau-t}e^{\sigma(s- \tau)}\bigl( \bigl\vert \omega(s) \bigr\vert ^{2}+ \bigl\vert \omega(s) \bigr\vert ^{p+1}\bigr)\,ds \\& \qquad {}+c \int^{\tau}_{\tau-t} \int_{ \vert x \vert \geq k}e^{\sigma(s-\tau)}g^{2}(x,s)\,dx\,ds. \end{aligned}$$
(4.50)
Replacing ω by \(\theta_{-\tau}\omega\), it then follows from (4.50) that
$$\begin{aligned}& \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \bigl( \bigl\vert v(\tau,\tau-t, \theta_{-\tau}\omega,v_{0}) \bigr\vert ^{2}+\bigl(\lambda+\delta^{2}- \beta_{2} \delta\bigr) \bigl\vert u(\tau,\tau-t, \theta_{-\tau}\omega,u_{0}) \bigr\vert ^{2} \\& \qquad {} + \bigl\vert \Delta u(\tau,\tau-t,\theta_{-\tau} \omega,u_{0}) \bigr\vert ^{2}+2F\bigl(x,u( \tau,\tau-t, \theta_{-\tau}\omega,u_{0})\bigr)\bigr)\,dx \\& \quad \leq c\eta+ e^{-\sigma t} \int_{\mathbb{R}^{n}}\rho\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \bigl( \vert v_{0} \vert ^{2}+\bigl(\lambda+ \delta^{2}-\beta_{2} \delta\bigr) \vert u_{0} \vert ^{2} + \vert \Delta u_{0} \vert ^{2}+2F(x,u_{0})\bigr)\,dx \\& \qquad {} +\frac{2\mu_{1}+8\mu_{2}}{k^{2}} \int^{\tau}_{\tau-t}e^{\sigma(s- \tau)}\bigl( \bigl\vert \Delta u(s,\tau-t,\theta_{-\tau}\omega,u_{0}) \bigr\vert ^{2}+ \bigl\vert v(s, \tau-t,\theta_{-\tau} \omega,v_{0}) \bigr\vert ^{2}\bigr)\,ds \\& \qquad {} +\frac{4\sqrt{2}\mu_{1}}{k} \int^{\tau}_{\tau-t}e^{\sigma(s-\tau)}\bigl( \bigl\vert \Delta u(s,\tau-t,\theta_{-\tau}\omega,u_{0}) \bigr\vert ^{2}+ \bigl\vert \nabla v(s, \tau-t,\theta_{-\tau} \omega,v_{0}) \bigr\vert ^{2}\bigr)\,ds \\& \qquad {} +c\eta \int^{\tau}_{\tau-t}e^{\sigma(s-\tau)}\bigl( \bigl\vert \theta_{-\tau} \omega(s) \bigr\vert ^{2}+ \bigl\vert \theta_{-\tau}\omega(s) \bigr\vert ^{p+1}\bigr) \,ds+c \int^{\tau}_{ \tau-t} \int_{ \vert x \vert \geq k}e^{\sigma(s-\tau)}g^{2}(x,s)\,dx\,ds \\& \quad \leq c\eta+ e^{-\sigma t} \int_{\mathbb{R}^{n}}\rho\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \bigl( \vert v_{0} \vert ^{2}+\bigl(\lambda+ \delta^{2}-\beta_{2} \delta\bigr) \vert u_{0} \vert ^{2} + \vert \Delta u_{0} \vert ^{2}+2F(x,u_{0})\bigr)\,dx \\& \qquad {} +\frac{2\mu_{1}+8\mu_{2}}{k^{2}} \int^{\tau}_{\tau-t}e^{\sigma(s- \tau)}\bigl( \bigl\vert \Delta u(s,\tau-t,\theta_{-\tau}\omega,u_{0}) \bigr\vert ^{2}+ \bigl\vert v(s, \tau-t,\theta_{-\tau} \omega,v_{0}) \bigr\vert ^{2}\bigr)\,ds \\& \qquad {} +\frac{4\sqrt{2}\mu_{1}}{k} \int^{\tau}_{\tau-t}e^{\sigma(s-\tau)}\bigl( \bigl\vert \Delta u(s,\tau-t,\theta_{-\tau}\omega,u_{0}) \bigr\vert ^{2}+ \bigl\vert \nabla v(s, \tau-t,\theta_{-\tau} \omega,v_{0}) \bigr\vert ^{2}\bigr)\,ds \\& \qquad {} +c\eta \int^{0}_{-\infty}e^{\sigma s}\bigl( \bigl\vert \omega(s) \bigr\vert ^{2}+ \bigl\vert \omega (s) \bigr\vert ^{p+1}\bigr)\,ds+c \int^{\tau}_{-\infty} \int_{ \vert x \vert \geq k}e^{\sigma(s-\tau)}g^{2}(x,s)\,dx\,ds. \end{aligned}$$
(4.51)
By (3.17), we see that there exists \(K_{2}=K_{2}(\tau,\eta)\geq K_{1}\) such that, for all \(k\geq K_{2}\),
$$ c \int^{\tau}_{-\infty} \int_{|x|\geq k}e^{\sigma(s-\tau)}g^{2}(x,s)\,dx\,ds \leq\eta. $$
(4.52)
It follows from (4.51)–(4.52), Lemma 4.1, and Lemma 4.2 that there exists \(T_{1}=T_{1}(\tau,\omega, D,\eta)>0\) such that, for all \(t\geq T_{1}\), \(k\geq K_{2}\),
$$\begin{aligned}& \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \bigl( \bigl\vert v(\tau,\tau-t, \theta_{-\tau}\omega,v_{0}) \bigr\vert ^{2}+\bigl(\lambda+\delta^{2}- \beta_{2} \delta\bigr) \bigl\vert u(\tau,\tau-t, \theta_{-\tau}\omega,u_{0}) \bigr\vert ^{2} \\& \qquad {} + \bigl\vert \Delta u(\tau,\tau-t,\theta_{-\tau} \omega,u_{0}) \bigr\vert ^{2}+2F\bigl(x,u( \tau,\tau-t, \theta_{-\tau}\omega,u_{0})\bigr)\bigr)\,dx \\& \quad \leq c\eta\biggl(1+ \int^{0}_{-\infty}e^{\sigma s}\bigl( \bigl\vert \omega(s) \bigr\vert ^{2}+ \bigl\vert \omega(s) \bigr\vert ^{p+1}\bigr)\,ds+ \int^{\tau}_{-\infty} \int_{ \vert x \vert \geq k}e^{ \sigma(s-\tau)}g^{2}(x,s)\,dx\,ds \biggr), \end{aligned}$$
(4.53)
where \((u_{0},v_{0})^{\top}\in D(\tau-t,\theta_{-t}\omega)\).
Note that (3.6) holds, then we find
$$ -2 \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)F(x,u)\,dx \leq2 \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \eta_{3}\,dx\leq2 \int_{ \vert x \vert \geq k}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \eta_{3}\,dx, $$
from which along with \(\eta_{3}\in L^{1}(\mathbb{R}^{n})\), we see that there exists \(K_{3}=K_{3}(\tau,\eta)\geq K_{2}\) such that, for all \(k\geq K_{3}\),
$$ -2 \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)F(x,u)\,dx \leq \eta. $$
(4.54)
Then from (4.53)–(4.54) we know that there exists \(T_{2}=T_{2}(\tau,\omega,D,\eta)>T_{1}\) such that, for all \(t\geq T_{2}\) and \(k\geq K_{3}\),
$$\begin{aligned}& \int_{\mathbb{R}^{n}}\rho\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \bigl( \bigl\vert v(\tau,\tau-t, \theta_{-\tau}\omega,v_{0}) \bigr\vert ^{2}+\bigl(\lambda+\delta^{2}- \beta_{2} \delta\bigr) \bigl\vert u(\tau,\tau-t, \theta_{-\tau}\omega,u_{0}) \bigr\vert ^{2} \\& \qquad {} + \bigl\vert \Delta u(\tau,\tau-t,\theta_{-\tau} \omega,u_{0}) \bigr\vert ^{2} \bigr)\,dx \\& \quad \leq c\eta\biggl(1+ \int^{0}_{-\infty}e^{\sigma s}\bigl( \bigl\vert \omega(s) \bigr\vert ^{2}+ \bigl\vert \omega(s) \bigr\vert ^{p+1}\bigr)\,ds+ \int^{\tau}_{-\infty} \int_{ \vert x \vert \geq k}e^{ \sigma(s-\tau)}g^{2}(x,s)\,dx\,ds \biggr), \end{aligned}$$
(4.55)
which completes the proof. □
Let \(\widehat{\rho}=1-\rho\) with ρ given by (4.36). Fix \(k\geq1\), and set
$$ \textstyle\begin{cases} \widehat{u}(t,\tau,\omega,\widehat{u_{0}})=\widehat{\rho}( \frac{ \vert x \vert ^{2}}{k^{2}})u(t,\tau,\omega,u_{0}), \\ \widehat{v}(t,\tau,\omega,\widehat{v_{0}})=\widehat{\rho}( \frac{ \vert x \vert ^{2}}{k^{2}})v(t,\tau,\omega,v_{0}). \end{cases} $$
(4.56)
By (3.10)–(3.11) we find that û and v̂ satisfy the following system in \(\mathbb{B}_{2k}=\{x\in\mathbb{R}^{n}:|x|<2k\}\):
$$\begin{aligned}& \frac{d\widehat{u}}{dt}=\widehat{v}+ \widehat{\rho}\biggl( \frac{ \vert x \vert ^{2}}{k^{2}} \biggr)\phi\omega(t)-\delta\widehat{u}, \end{aligned}$$
(4.57)
$$\begin{aligned}& \frac{d\widehat{v}}{dt}-\delta\widehat{v}+\bigl(\delta^{2}+\lambda+A \bigr) \widehat{u}+\widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)f(x,u) \\& \quad = \widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)g(x,t)-\widehat{\rho} \biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)h\bigl(v+ \phi\omega(t) -\delta u\bigr)+(1+ \delta) \widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\phi\omega(t) \\& \qquad {} +4\Delta\nabla\widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \nabla u+6 \Delta\widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \Delta u \\& \qquad {}+4\nabla \widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \Delta\nabla u+u \Delta^{2} \widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr), \end{aligned}$$
(4.58)
with boundary conditions
$$ \widehat{u}=\widehat{v}=0 \quad \text{for } \vert x \vert =2k. $$
(4.59)
Let \(\{e_{n}\}^{\infty}_{n=1}\) be an orthonormal basis of \(L^{2}(\mathbb{B}_{2k})\) such that \(Ae_{n}=\lambda_{n}e_{n}\) with zero boundary condition in \(\mathbb{B}_{2k}\). Given n, let \(X_{n}=\operatorname{span}\{e_{1},\ldots,e_{n}\}\) and \(P_{n}:L^{2}(\mathbb{B}_{2k})\rightarrow X_{n}\) be the projection operator.
Lemma 4.4
Under (3.4)–(3.9) and (3.14)–(3.17), for every\(\eta>0\), \(\tau\in\mathbb{R}\), \(\omega\in\varOmega\), \(D=\{D(\tau,\omega):\tau\in\mathbb{R},\omega\in\varOmega\}\in \mathcal{D}\), there exist\(T=T(\tau,\omega,D,\eta)>0\), \(K=K(\tau,\omega,\eta)\geq1\), and\(N=N(\tau,\omega,\eta)\geq1\)such that, for all\(t\geq T\), \(k\geq K\), and\(n\geq N\), the solution of problem (4.57)–(4.59) satisfies
$$ \bigl\Vert (I-P_{n})\widehat{Y}\bigl(\tau,\tau-t, \theta_{-\tau}\omega,D(\tau-t, \theta_{-\tau}\omega)\bigr) \bigr\Vert ^{2}_{E (\mathbb{B}_{2k})}\leq\eta. $$
Proof
Let \(\widehat{u}_{n,1}=P_{n}\widehat{u}\), \(\widehat{u}_{n,2}=(I-P_{n}) \widehat{u}\), \(\widehat{v}_{n,1}=P_{n}\widehat{v}\), \(\widehat{v}_{n,2}=(I-P_{n}) \widehat{v}\). Applying \(I-P_{n}\) to (4.57), we obtain
$$ \widehat{v}_{n,2}=\frac{d\widehat{u}_{n,2}}{dt}+\delta\widehat {u}_{n,2}-(I-P_{n}) \widehat{\rho}\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\phi\omega(t). $$
(4.60)
Applying \(I-P_{n}\) to (4.58) and taking the inner product with \(\widehat{v}_{n,2}\) in \(L^{2}(\mathbb{B}_{2k})\), we have
$$\begin{aligned}& \frac{1}{2}\frac{d}{dt} \Vert \widehat{v}_{n,2} \Vert ^{2}- \delta \Vert \widehat{v}_{n,2} \Vert ^{2} +\bigl(\lambda+\delta^{2}+A\bigr) ( \widehat{u}_{n,2}, \widehat{v}_{n,2}) +\biggl( \widehat{ \rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)f(x,u), \widehat{v}_{n,2} \biggr) \\& \quad = \biggl( (I-P_{n}) \widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}} \biggr)g(x,t), \widehat{v}_{n,2}\biggr)+(1+\delta) \biggl(\widehat{ \rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \phi\omega(t),\widehat{v}_{n,2} \biggr) \\& \qquad {} - (I-P_{n})\widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}} \biggr) \bigl(h\bigl(v+\phi\omega(t)- \delta u\bigr),\widehat{v}_{n,2} \bigr) \\& \qquad {} + \biggl(4\Delta\nabla\widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}} \biggr) \nabla u+6 \Delta\widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \Delta u \\& \qquad {}+4\nabla \widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \Delta\nabla u+u \Delta^{2} \widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr), \widehat{v}_{n,2}\biggr). \end{aligned}$$
(4.61)
Substituting \(\widehat{v}_{n,2}\) in (4.60) into the third term on the left-hand side of (4.61), we have
$$\begin{aligned} \bigl(\lambda+\delta^{2}\bigr) (\widehat{u}_{n,2}, \widehat{v}_{n,2}) =&\biggl( \widehat{u}_{n,2}, \frac{d\widehat{u}_{n,2}}{dt}+\delta\widehat {u}_{n,2}-(I-P_{n}) \widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\phi\omega(t)\biggr) \\ \geq&\frac{1}{2}\bigl(\lambda+\delta^{2}\bigr) \frac{d}{dt} \Vert \widehat{u}_{n,2} \Vert ^{2}+\delta\bigl(\lambda+\delta^{2}\bigr) \Vert \widehat{u}_{n,2} \Vert ^{2} -\frac{1}{4}\delta\bigl(\lambda+\delta^{2}- \beta_{2}\delta\bigr) \Vert \widehat{u}_{n,2} \Vert ^{2} \\ &{}-c \biggl\Vert (I-P_{n}) \widehat{\rho}\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\phi \biggr\Vert ^{2} \bigl\vert \omega(t) \bigr\vert ^{2}, \end{aligned}$$
(4.62)
and then
$$\begin{aligned} (A\widehat{u}_{n,2},\widehat{v}_{n,2}) =&\bigg(\Delta \widehat{u}_{n,2}, \Delta\biggl(\frac{d\widehat{u}_{n,2}}{dt}+\delta \widehat{u}_{n,2} -(I-P_{n}) \widehat{\rho}\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\phi\omega(t)\biggr)\bigg) \\ \geq&\frac{1}{2}\frac{d}{dt} \Vert \Delta \widehat{u}_{n,2} \Vert ^{2}+ \frac{3\delta}{4} \Vert \Delta\widehat{u}_{n,2} \Vert ^{2} \\ &{}-c \biggl\Vert (I-P_{n}) \Delta\biggl( \widehat{\rho}\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\phi\biggr) \biggr\Vert ^{2} \bigl\vert \omega(t) \bigr\vert ^{2}. \end{aligned}$$
(4.63)
For the fourth term on the left-hand side of (4.61), we have
$$\begin{aligned}& \biggl( \widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)f(x,u), \widehat{v}_{n,2}\biggr) \\& \quad = \biggl( \widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)f(x,u), \frac{d\widehat{u}_{n,2}}{dt}+\delta\widehat{u}_{n,2}-(I-P_{n}) \widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\phi\omega(t)\biggr) \\& \quad = \frac{d}{dt}\biggl( \widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}} \biggr)f(x,u), \widehat{u}_{n,2}\biggr)- \biggl( \widehat{\rho}\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)f'_{u}(x,u)u_{t}, \widehat{u}_{n,2}\biggr) \\& \qquad {} + \delta\biggl(\widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}} \biggr)f(x,u),\widehat{u}_{n,2}\biggr)-\biggl( \widehat{\rho}\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)f(x,u),(I-P_{n}) \widehat{\rho}\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\phi\omega(t)\biggr). \end{aligned}$$
(4.64)
For the third term on the right-hand side of (4.61), we have
$$\begin{aligned}& -(I-P_{n})\widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \bigl(h \bigl(v+\phi\omega(t)- \delta u\bigr),\widehat{v}_{n,2}\bigr) \\& \quad = -(I-P_{n})\widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \bigl(h\bigl(v+\phi\omega(t)- \delta u\bigr)-h(0),\widehat{v}_{n,2} \bigr) \\& \quad \leq -\beta_{1} \Vert \widehat{v}_{n,2} \Vert ^{2}+h'(\vartheta)\delta( \widehat{u}_{n,2}, \widehat{v}_{n,2})+\frac{\beta_{1}-\delta}{6} \Vert \widehat{v}_{n,2} \Vert ^{2}+c \biggl\Vert (I-P_{n}) \widehat{\rho} \biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\phi \biggr\Vert ^{2} \bigl\vert \omega(t) \bigr\vert ^{2} \\& \quad \leq -\beta_{1} \Vert \widehat{v}_{n,2} \Vert ^{2}+h'(\vartheta)\delta\biggl( \widehat{u}_{n,2}, \frac{d\widehat{u}_{n,2}}{dt}+\delta\widehat {u}_{n,2}-(I-P_{n}) \widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\phi\omega(t)\biggr) \\& \qquad {} + \frac{\beta_{1}-\delta}{6} \Vert \widehat{v}_{n,2} \Vert ^{2}+c \biggl\Vert (I-P_{n}) \widehat{\rho}\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\phi \biggr\Vert ^{2} \bigl\vert \omega(t) \bigr\vert ^{2} \\& \quad \leq -\beta_{1} \Vert \widehat{v}_{n,2} \Vert ^{2}+\beta_{2}\delta\cdot \frac{1}{2} \frac{d}{dt} \Vert \widehat{u}_{n,2} \Vert ^{2}+ \beta_{2}\delta^{2} \Vert \widehat{u}_{n,2} \Vert ^{2} \\& \qquad {}+\beta_{2}\delta \biggl\Vert (I-P_{n}) \widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}} \biggr)\phi \biggr\Vert \bigl\vert \omega(t) \bigr\vert \Vert \widehat{u}_{n,2} \Vert \\& \qquad {} +\frac{\beta_{1}-\delta}{6} \Vert \widehat{v}_{n,2} \Vert ^{2}+c \biggl\Vert (I-P_{n}) \widehat{\rho}\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\phi \biggr\Vert ^{2} \bigl\vert \omega(t) \bigr\vert ^{2} \\& \quad \leq -\beta_{1} \Vert \widehat{v}_{n,2} \Vert ^{2}+\beta_{2}\delta\cdot \frac{1}{2} \frac{d}{dt} \Vert \widehat{u}_{n,2} \Vert ^{2}+ \beta_{2}\delta^{2} \Vert \widehat{u}_{n,2} \Vert ^{2}+\frac{1}{4}\delta \bigl(\lambda+\delta^{2}- \beta_{2}\delta\bigr) \Vert \widehat{u}_{n,2} \Vert ^{2} \\& \qquad {} +\frac{\beta_{1}-\delta}{6} \Vert \widehat{v}_{n,2} \Vert ^{2}+c \biggl\Vert (I-P_{n}) \widehat{\rho}\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\phi \biggr\Vert ^{2} \bigl\vert \omega(t) \bigr\vert ^{2}. \end{aligned}$$
(4.65)
Using the Cauchy–Schwarz inequality and Young’s inequality, we get
$$\begin{aligned}& \delta\biggl(\widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\phi\omega(t), \widehat{v}_{n,2}\biggr) \\& \quad \leq \frac{\beta_{1}-\delta}{6} \Vert \widehat{v}_{n,2} \Vert ^{2}+c \biggl\Vert (I-P_{n}) \widehat{\rho}\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\phi \biggr\Vert ^{2} \bigl\vert \omega(t) \bigr\vert ^{2}, \end{aligned}$$
(4.66)
$$\begin{aligned}& \biggl((I-P_{n})\widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}} \biggr)g(x,t),\widehat{v}_{n,2}\biggr) \\& \quad \leq \frac{\beta_{1}-\delta}{6} \Vert \widehat{v}_{n,2} \Vert ^{2}+c \biggl\Vert (I-P_{n}) \biggl( \widehat{\rho} \biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)g(x,t)\biggr) \biggr\Vert ^{2}. \end{aligned}$$
(4.67)
Now, we estimate the last term in (4.61)
$$\begin{aligned}& \biggl(4\Delta\nabla\widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\cdot\nabla u+6 \Delta\widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\cdot\Delta u+4\nabla \widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\cdot\Delta\nabla u+u \Delta^{2} \widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr), \widehat{v}_{n,2}\biggr) \\& \quad = \biggl(4\nabla u\cdot\biggl(\frac{12 \vert x \vert }{k^{4}}\widehat{\rho} ''\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)+ \frac{8 \vert x \vert ^{3}}{k^{6}} \widehat{\rho} ''' \biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\biggr) \\& \qquad {}+ 6\Delta u\cdot\biggl( \frac{2}{k^{2}} \widehat{\rho} '\biggl(\frac{ \vert x \vert ^{2}}{r^{2}} \biggr)+\frac{4x^{2}}{k^{4}} \widehat{\rho} '' \biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\biggr)+\frac{8 \vert x \vert }{k^{2}}\Delta\nabla u\cdot\widehat{\rho} '\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr) \\& \qquad {}+u\biggl( \frac{12}{k^{4}}\widehat{\rho} ''\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)+\frac{48x^{2}}{k^{6}}\widehat{\rho} ''' \biggl( \frac{ \vert x \vert ^{2}}{k^{2}} \biggr) +\frac{16x^{4}}{k^{8}}\widehat{\rho} '''' \biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\biggr),\widehat{v}_{n,2}\biggr) \\& \quad \leq \frac{16\sqrt{2}(3\mu_{2}+4\mu_{3})}{k^{3}} \Vert \nabla u \Vert \cdot \Vert \widehat{v}_{n,2} \Vert +\frac{12(\mu_{1}+4\mu_{2})}{k^{2}} \Vert \Delta u \Vert \cdot \Vert \widehat{v}_{n,2} \Vert \\& \qquad {} +\frac{8\sqrt{2}\mu_{1}}{k} \bigl\Vert A^{\frac{3}{4}} u \bigr\Vert \cdot \Vert \widehat{v}_{n,2} \Vert +\frac{4(3\mu_{2}+24\mu_{3}+16\mu_{4})}{k^{4}} \Vert u \Vert \cdot \Vert \widehat{v}_{n,2} \Vert \\& \quad \leq \frac{8(48\mu_{2}+64\mu_{3})^{2}}{(\beta_{1}-\delta) k^{6}} \Vert \nabla u \Vert ^{2}+ \frac{4(12\mu_{1}+48\mu_{2})^{2}}{(\beta_{1}-\delta)k^{4}} \Vert \Delta u \Vert ^{2} + \frac{512\mu^{2}_{1}}{(\beta_{1}-\delta)k^{2}} \bigl\Vert A^{ \frac{3}{4}} u \bigr\Vert ^{2} \\& \qquad {} + \frac{4(12\mu_{2}+96\mu_{3}+64\mu_{4})^{2}}{(\beta_{1}-\delta)k^{8}} \Vert u \Vert ^{2} + \frac{\beta_{1}-\delta}{4} \Vert \widehat{v}_{n,2} \Vert ^{2}. \end{aligned}$$
(4.68)
Assemble together (4.61)–(4.68) to obtain
$$\begin{aligned}& \frac{1}{2}\frac{d}{dt}\biggl[ \Vert \widehat{v}_{n,2} \Vert ^{2}+\bigl(\lambda+\delta^{2}- \beta_{2}\delta\bigr) \Vert \widehat{u}_{n,2} \Vert ^{2}+ \Vert \Delta\widehat{u}_{n,2} \Vert ^{2}+2\biggl( \widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}} \biggr)f(x,u),\widehat{u}_{n,2}\biggr)\biggr] \\& \qquad {} +\delta\bigl[ \Vert \widehat{v}_{n,2} \Vert ^{2}+\bigl(\lambda+\delta^{2}-\beta_{2} \delta\bigr) \Vert \widehat{u}_{n,2} \Vert ^{2}+ \Vert \Delta\widehat{u}_{n,2} \Vert ^{2}\bigr]+ \delta \biggl( \widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)f(x,u), \widehat{u}_{n,2}\biggr) \\& \quad \leq \frac{\delta}{2} \Vert \widehat{v}_{n,2} \Vert ^{2}+ \frac{3\delta-\beta_{1}}{4} \Vert \widehat{v}_{n,2} \Vert ^{2}+ \frac{\delta}{2} \bigl(\lambda+ \delta^{2}-\beta_{2}\delta\bigr) \Vert \widehat{u}_{n,2} \Vert ^{2}+\frac{\delta}{4} \Vert \Delta\widehat{u}_{n,2} \Vert ^{2} \\& \qquad {} +\frac{2}{\beta_{1}-\delta}\biggl( \frac{4(48\mu_{2}+64\mu_{3})^{2}}{ k^{6}} \Vert \nabla u \Vert ^{2}+ \frac{2(12\mu_{1}+48\mu_{2})^{2}}{k^{4}} \Vert \Delta u \Vert ^{2}+ \frac{256\mu^{2}_{1}}{k^{2}} \bigl\Vert A^{\frac{3}{4}} u \bigr\Vert ^{2} \\& \qquad {} +\frac{2(12\mu_{2}+96\mu_{3}+64\mu_{4})^{2}}{k^{8}} \Vert u \Vert ^{2}\biggr)+c \biggl\Vert (I-P_{n}) \biggl(\widehat{\rho}\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)g(x,t)\biggr) \biggr\Vert ^{2} \\& \qquad {} +c \biggl\Vert (I-P_{n}) \widehat{\rho}\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\phi \biggr\Vert ^{2} \bigl\vert \omega(t) \bigr\vert ^{2}+ c \biggl\Vert (I-P_{n}) \Delta\biggl( \widehat{\rho}\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\phi\biggr) \biggr\Vert ^{2} \bigl\vert \omega(t) \bigr\vert ^{2} \\& \qquad {} +\biggl( \widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}} \biggr)f'_{u}(x,u)u_{t}, \widehat{u}_{n,2}\biggr)+\biggl( \widehat{\rho}\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)f(x,u),(I-P_{n}) \widehat{\rho}\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\phi\omega(t)\biggr). \end{aligned}$$
(4.69)
For the nonlinear terms in (4.69), by (3.7), using Cauchy’s inequality and Young’s inequality, we obtain
$$ \biggl( \widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)f'_{u}(x,u)u_{t}, \widehat{u}_{n,2} \biggr) \leq\frac{\delta}{4} \Vert \Delta \widehat{u}_{n,2} \Vert ^{2}+c\lambda^{-1}_{n+1} \Vert u_{t} \Vert ^{2}. $$
(4.70)
By (3.4), we know
$$\begin{aligned}& \biggl( \widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)f(x,u),(I-P_{n}) \widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\phi\omega(t)\biggr) \\& \quad \leq c \biggl\Vert (I-P_{n}) \widehat{\rho}\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\phi \biggr\Vert \bigl\vert \omega(t) \bigr\vert +c \Vert u \Vert ^{ p }_{H^{2}(\mathbb{R}^{n})} \biggl\Vert (I-P_{n}) \widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\phi \biggr\Vert \bigl\vert \omega(t) \bigr\vert . \end{aligned}$$
(4.71)
Since \(1\leq p\leq\frac{n+4}{n-4}\) and \(\lambda_{n}\rightarrow\infty\), by Lemmas 4.1 and 4.2, there are \(N_{1}=N(\eta)\), \(K_{1}=K(\eta)\) such that, for all \(n\geq N_{1}\), \(k\geq K_{1}\),
$$\begin{aligned}& \frac{2}{\beta_{1}-\delta}\biggl( \frac{4(48\mu_{2}+64\mu_{3})^{2}}{ k^{6}} \Vert \nabla u \Vert ^{2}+ \frac{2(12\mu_{1}+48\mu_{2})^{2}}{k^{4}} \Vert \Delta u \Vert ^{2}+ \frac{256\mu^{2}_{1}}{k^{2}} \bigl\Vert A^{\frac{3}{4}} u \bigr\Vert ^{2} \\& \qquad {} +\frac{2(12\mu_{2}+96\mu_{3}+64\mu_{4})^{2}}{k^{8}} \Vert u \Vert ^{2}\biggr)+c \lambda^{- \frac{1}{2}}_{n+1} \Vert u_{t} \Vert ^{2} \\& \qquad {} +c \biggl\Vert (I-P_{n}) \widehat{\rho}\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\phi \biggr\Vert \bigl\vert \omega(t) \bigr\vert +c \Vert u \Vert ^{ p }_{H^{2}(\mathbb{R}^{n})} \biggl\Vert (I-P_{n}) \widehat{\rho}\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\phi \biggr\Vert \bigl\vert \omega(t) \bigr\vert \\& \qquad {} +c \biggl\Vert (I-P_{n}) \widehat{\rho}\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\phi \biggr\Vert ^{2} \bigl\vert \omega(t) \bigr\vert ^{2} +c \biggl\Vert (I-P_{n}) \Delta\biggl( \widehat{\rho}\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)\phi\biggr) \biggr\Vert ^{2} \bigl\vert \omega(t) \bigr\vert ^{2} \\& \quad \leq c\eta\bigl(1+ \bigl\vert \omega(t) \bigr\vert ^{2}+ \Vert u_{t} \Vert ^{2}+ \Vert u \Vert ^{18}_{H^{2}( \mathbb{R}^{n})}\bigr). \end{aligned}$$
(4.72)
Then, by (4.69)–(4.72), we obtain
$$\begin{aligned}& \frac{d}{dt}\biggl[ \Vert \widehat{v}_{n,2} \Vert ^{2}+\bigl(\lambda+\delta^{2}-\beta_{2} \delta\bigr) \Vert \widehat{u}_{n,2} \Vert ^{2}+ \Vert \Delta\widehat{u}_{n,2} \Vert ^{2}+2\biggl( \widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)f(x,u),\widehat{u}_{n,2} \biggr)\biggr] \\& \qquad {} +\sigma\biggl[ \Vert \widehat{v}_{n,2} \Vert ^{2}+\bigl(\lambda+\delta^{2}-\beta_{2} \delta\bigr) \Vert \widehat{u}_{n,2} \Vert ^{2}+ \Vert \Delta\widehat{u}_{n,2} \Vert ^{2}+2\biggl( \widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)f(x,u),\widehat{u}_{n,2} \biggr)\biggr] \\& \quad \leq c\eta\bigl(1+ \bigl\vert \omega(t) \bigr\vert ^{2}+ \Vert u_{t} \Vert ^{2}+ \Vert u \Vert ^{18}_{H^{2}( \mathbb{R}^{n})}\bigr)+c \biggl\Vert (I-P_{n}) \biggl(\widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)g(x,t)\biggr) \biggr\Vert ^{2}. \end{aligned}$$
(4.73)
Multiplying (4.73) by \(e^{\sigma t}\) and then integrating over \((\tau-t,\tau)\), we have for all \(n>N_{1}\) and \(k>K_{1}\)
$$\begin{aligned}& \bigl\Vert \widehat{v}_{n,2}(\tau,\tau-t,\omega) \bigr\Vert ^{2}+\bigl(\lambda+\delta^{2}- \beta_{2} \delta\bigr) \bigl\Vert \widehat{u}_{n,2}(\tau,\tau-t,\omega) \bigr\Vert ^{2} + \bigl\Vert \Delta\widehat{u}_{n,2}(\tau, \tau-t,\omega) \bigr\Vert ^{2} \\& \qquad {}+2\biggl(\widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)f(x,u), \widehat{u}_{n,2}( \tau,\tau-t,\omega)\biggr) \\& \quad \leq e^{-\sigma t} \bigg( \Vert v_{0} \Vert ^{2}+ \Vert u_{0} \Vert ^{2}+\bigl( \lambda+\delta^{2}- \beta_{2}\delta\bigr) \Vert \Delta u_{0} \Vert ^{2}+2\biggl(\widehat{\rho}\biggl( \frac{ \vert x \vert ^{2}}{k^{2}}\biggr)f(x,u),u_{0}\biggr)\bigg) \\& \qquad {}+c\eta \int^{\tau}_{\tau-t}e^{\sigma(s-\tau)}\bigl(1+ \bigl\vert \omega(s) \bigr\vert ^{2}+ \bigl\Vert u_{t}(s, \tau-t,\omega,u_{0}) \bigr\Vert ^{2} + \bigl\Vert u(s,\tau-t,\omega,u_{0}) \bigr\Vert ^{18}_{H^{2}(\mathbb{R}^{n})} \bigr)\,ds \\& \qquad {}+c \int^{\tau}_{\tau-t}e^{\sigma(s-\tau)} \biggl\Vert (I-P_{n}) \widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{r^{2}}\biggr)g(x,s) \biggr\Vert ^{2}\,ds. \end{aligned}$$
(4.74)
Replacing ω by \(\theta_{-\tau}\omega\), by a similar process as in Lemma 4.1, we get
$$\begin{aligned}& \bigl\Vert \widehat{v}_{n,2}(\tau,\tau-t,\theta_{-\tau} \omega) \bigr\Vert ^{2}+\bigl( \lambda+\delta^{2}- \beta_{2}\delta\bigr) \bigl\Vert \widehat{u}_{n,2}(\tau, \tau-t,\theta_{-\tau}\omega) \bigr\Vert ^{2} \\& \qquad {} + \bigl\Vert \Delta\widehat{u}_{n,2}(\tau,\tau-t, \theta_{-\tau}\omega) \bigr\Vert ^{2} +2\biggl(\widehat{ \rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)f(x,u),\widehat{u}_{n,2}( \tau, \tau-t,\theta_{-\tau}\omega)\biggr) \\& \quad \leq ce^{-\sigma t} \bigl(1+ \Vert v_{0} \Vert ^{2}+ \Vert u_{0} \Vert ^{2}+\bigl( \lambda+ \delta^{2}-\beta_{2}\delta\bigr) \Vert \Delta u_{0} \Vert ^{2}+ \Vert u_{0} \Vert ^{2}_{H^{2}( \mathbb{R}^{n})}+ \Vert u_{0} \Vert ^{p+1}_{H^{2}(\mathbb{R}^{n})}\bigr) \\& \qquad {} +c\eta \int^{\tau}_{\tau-t}e^{\sigma(s-\tau)}\bigl(1+ \bigl\vert \theta_{-\tau} \omega(s) \bigr\vert ^{2}+ \bigl\Vert u_{t}(s,\tau-t,\theta_{-\tau}\omega,u_{0}) \bigr\Vert ^{2} \\& \qquad {} + \bigl\Vert u(s,\tau-t,\theta_{-\tau}\omega,u_{0}) \bigr\Vert ^{18}_{H^{2}( \mathbb{R}^{n})}\bigr)\,ds+c \int^{\tau}_{\tau-t}e^{\sigma(s-\tau)} \biggl\Vert (I-P_{n}) \widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{r^{2}}\biggr)g(x,s) \biggr\Vert ^{2}\,ds \\& \quad \leq c e^{-\sigma t} \bigl(1+ \Vert v_{0} \Vert ^{2}+ \Vert u_{0} \Vert ^{2}+\bigl( \lambda+ \delta^{2}-\beta_{2}\delta\bigr) \Vert \Delta u_{0} \Vert ^{2}+ \Vert u_{0} \Vert ^{2}_{H^{2}( \mathbb{R}^{n})}+ \Vert u_{0} \Vert ^{p+1}_{H^{2}(\mathbb{R}^{n})}\bigr) \\& \qquad {} +c\eta \int^{0}_{-\infty}e^{\sigma s }\bigl(1+ \bigl\vert \omega(s) \bigr\vert ^{2}\bigr)\,ds+c \eta \int^{\tau}_{\tau-t}e^{\sigma(s-\tau)}\bigl( \bigl\Vert u_{t}(s,\tau-t, \theta_{-\tau}\omega,u_{0}) \bigr\Vert ^{2} \\& \qquad {} + \bigl\Vert u(s,\tau-t,\theta_{-\tau}\omega,u_{0}) \bigr\Vert ^{18}_{H^{2}( \mathbb{R}^{n})}\bigr)\,ds \\& \qquad {}+c \int^{0}_{-\infty}e^{\sigma s } \biggl\Vert (I-P_{n}) \widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{r^{2}}\biggr)g(x,s+ \tau) \biggr\Vert ^{2}\,ds. \end{aligned}$$
(4.75)
Using the first equation of (3.10) as well as the Minkowski inequality, we can obtain
$$\begin{aligned}& \bigl\Vert u_{t}(s,\tau-t,\theta_{-\tau} \omega,u_{0}) \bigr\Vert ^{2} \\& \quad = \bigl\Vert -\delta u(s,\tau-t,\theta_{-\tau} \omega,u_{0})+v(s,\tau-t, \theta_{-\tau} \omega,v_{0}) +\phi\theta_{-\tau}\omega \bigr\Vert ^{2} \\& \quad \leq c\bigl( \bigl\Vert u(s,\tau-t,\theta_{-\tau} \omega,u_{0}) \bigr\Vert ^{2}+ \bigl\Vert v(s, \tau-t,\theta_{-\tau}\omega,v_{0}) \bigr\Vert ^{2}+ \vert \theta_{-\tau}\omega \vert ^{2} \bigr) \\& \quad \leq cR_{1}(\tau,\omega)+c \vert \theta_{-\tau} \omega \vert ^{2} \end{aligned}$$
(4.76)
and
$$ \bigl\Vert u(s,\tau-t,\theta_{-\tau}\omega,u_{0}) \bigr\Vert ^{18}_{H^{2}(\mathbb{R}^{n})} \leq cR^{9}_{1}( \tau,\omega), $$
(4.77)
where \(c=\max\{\delta,\|\phi\|^{2},1\}\) and \(R_{1}(\tau,\omega)\) is given in Lemma 4.1. Hence, it follows from (4.75)–(4.77) that
$$\begin{aligned}& \bigl\Vert \widehat{v}_{n,2}(\tau,\tau-t,\theta_{-\tau} \omega) \bigr\Vert ^{2}+\bigl( \lambda+\delta^{2}- \beta_{2}\delta\bigr) \bigl\Vert \widehat{u}_{n,2}(\tau, \tau-t,\theta_{-\tau}\omega) \bigr\Vert ^{2} \\& \qquad {} + \bigl\Vert \Delta\widehat{u}_{n,2}(\tau,\tau-t, \theta_{-\tau}\omega) \bigr\Vert ^{2} +2\biggl(\widehat{ \rho}\biggl(\frac{ \vert x \vert ^{2}}{k^{2}}\biggr)f(x,u),\widehat{u}_{n,2}( \tau, \tau-t,\theta_{-\tau}\omega)\biggr) \\& \quad \leq e^{-\sigma t} \bigl(1+ \Vert v_{0} \Vert ^{2}+ \Vert u_{0} \Vert ^{2}+\bigl( \lambda+\delta^{2}- \beta_{2}\delta\bigr) \Vert \Delta u_{0} \Vert ^{2}+ \Vert u_{0} \Vert ^{2}_{H^{2}( \mathbb{R}^{n})}+ \Vert u_{0} \Vert ^{p+1}_{H^{2}(\mathbb{R}^{n})}\bigr) \\& \qquad {} +c\eta R^{9}_{1}(\tau,\omega)+c\eta \int^{0}_{-\infty}e^{\sigma s }\bigl(1+ \bigl\vert \omega(s) \bigr\vert ^{2} \bigr)\,ds \\& \qquad {}+c \int^{0}_{-\infty}e^{\sigma s } \biggl\Vert (I-P_{n}) \widehat{\rho}\biggl(\frac{ \vert x \vert ^{2}}{r^{2}}\biggr)g(x,s+ \tau) \biggr\Vert ^{2}\,ds. \end{aligned}$$
(4.78)
Since \((u_{0},v_{0})^{\top}\in D(\tau-t,\theta_{-t}\omega)\) and \(D\in\mathcal{D}\), then
$$\begin{aligned}& e^{-\sigma t} \bigl(1+ \Vert v_{0} \Vert ^{2}+ \Vert u_{0} \Vert ^{2}+\bigl(\lambda+ \delta^{2}- \beta_{2}\delta\bigr) \Vert \Delta u_{0} \Vert ^{2}+ \Vert u_{0} \Vert ^{2}_{H^{2}( \mathbb{R}^{n})}+ \Vert u_{0} \Vert ^{p+1}_{H^{2}(\mathbb{R}^{n})}\bigr) \\& \quad \rightarrow0,\quad t\rightarrow\infty. \end{aligned}$$
(4.79)
For the last term on the right-hand side of (4.78), by (3.16), there exists \(N_{2}=N_{2}(\tau,\omega,\eta)\geq N_{1}\) such that, for all \(n\geq N_{2}\),
$$ \int^{0}_{-\infty} e^{\sigma s} \biggl\Vert (I-P_{n}) \biggl(\widehat{\rho}\biggl( \frac{ \vert x \vert ^{2}}{k^{2}} \biggr)g(x,s+\tau)\biggr) \biggr\Vert ^{2}\,ds < \eta. $$
(4.80)
The proof is completed by (3.4), (4.79)–(4.80), and Lemma 4.1. □