In this section, we will apply Proposition 2.8 to prove the existence of random attractors for the Klein-Gordon-Schrödinger system with ϵ-small random perturbation. The main tool is the tail-estimates method which is extensively used to prove the existence of random attractor, see [11, 12].
Consider the Klein-Gordon-Schrödinger system with ϵ-small random perturbation
$$ \begin{aligned} &i\,du+(\triangle u+i\alpha u+uv)\,dt=f\,dt+ \epsilon u\,dW_{1}, \quad x\in\mathbb{R}^{n}, t>0, \\ &dv_{t}+ \bigl(\nu v_{t}-\triangle v+\mu v- \beta|u|^{2} \bigr)\,dt=g\,dt+\epsilon\delta \,dW_{2}, \quad x\in \mathbb{R}^{n}, t>0, \end{aligned} $$
(3.1)
with the initial value conditions
$$ u(x,0)=u_{0}(x),\qquad v(x,0)=v_{0}(x), \qquad v_{t}(x,0)=v_{1}(x),\quad x\in\mathbb{R}^{n}, $$
where α, ϵ, ν, μ and β are positive constants, \(n\leq3\), \(f, g\in\mathbb{L}^{2}\), \(\delta\in\mathbb{H}^{1}\), \(W_{1}\) and \(W_{2}\) are independent two-side real-valued Wiener processes on a probability space \((\Omega,\mathbb{F},\mathbb{P})\).
For our purpose, we introduce the probability space
$$\Omega= \bigl\{ \omega=(\omega_{1},\omega_{2})\in\mathbb{C} \bigl(\mathbb{R},\mathbb {R}^{2} \bigr): \omega(0)=0 \bigr\} $$
endowed with the compact open topology [5]. Then we have \((W_{1}(t,\omega),W_{2}(t,\omega))=\omega(t)\), \(t\in\mathbb{R}\). Let ℙ be the corresponding Wiener measure, \(\mathbb{F}\) be the ℙ-completion of the Borel σ-algebra on Ω, and \(\theta_{t}\omega(\cdot)=\omega(\cdot+t)-\omega(t)\), \(t\in\mathbb{R}\). Then \((\Omega,\mathbb{F},\mathbb{P},(\theta_{t})_{t\in\mathbb{R}})\) is a metric dynamical system with the filtration \(\mathbb{F}_{t}:=\bigvee_{s\leq t}\mathbb{F}^{t}_{s}\), \(t\in\mathbb{R}\), where \(\mathbb{F}^{t}_{s}=\sigma\{W(t_{2})-W(t_{1}): s\le t_{1}\leq t_{2}\leq t\}\) is the smallest σ-algebra generated by the random variable \(W(t_{2})-W(t_{1})\) for all \(t_{1}\), \(t_{2}\) such that \(s\le t_{1}\leq t_{2}\leq t\), see [5] for more details.
We introduce an Ornstein-Uhlenbeck process \((\Omega,\mathbb{F},\mathbb{P},\theta_{t})\) given by the Wiener process:
$$\begin{aligned}& y_{1}(\theta_{t}\omega_{1})=-\nu\int _{-\infty}^{0}e^{\nu h}(\theta_{t} \omega _{1}) (h)\,dh, \quad t\in\mathbb{R}, \\& y_{2}(\theta_{t}\omega_{2})=-\lambda\int _{-\infty}^{0}e^{\lambda h}(\theta_{t} \omega_{2}) (h)\,dh,\quad t\in\mathbb{R}, \end{aligned}$$
where ν and λ are positive. The above integral exists in the sense that for any path ω with a subexponential growth, \(y_{1}\), \(y_{2}\) solve the following Itô equations:
$$\begin{aligned}& dy_{1}+\nu y_{1}\,dt=dW_{1}(t), \quad t\in \mathbb{R}, \\& dy_{2}+\lambda y_{2}\,dt=dW_{2}(t), \quad t\in \mathbb{R}. \end{aligned}$$
Furthermore, there exists a \(\theta_{t}\) invariant set \(\Omega'\subset\Omega\) of full ℙ measure such that:
-
(1)
the mappings \(t\rightarrow y_{i}(\theta_{t}\omega_{i})\), \(i=1,2\), are continuous for each \(\omega\in\Omega'\);
-
(2)
the random variables \(\|y_{i}(\omega_{i})\|\), \(i=1,2\), are tempered (for more details, see [11, 12]).
Let \(z_{1}(\theta_{t}\omega)=y_{1}(\theta_{t}\omega_{1})\) and \(z_{2}(\theta_{t}\omega)=\delta y_{2}(\theta_{t}\omega_{2})\). Then we have
$$\begin{aligned}& dz_{1}+\nu z_{1}\,dt=dW_{1}(t), \quad t\in \mathbb{R}, \\& dz_{2}+\lambda z_{2}\,dt=\delta\,dW_{2}(t),\quad t \in\mathbb{R}. \end{aligned}$$
Lemma 3.1
([23])
There exists a
\((\theta_{t})_{t\in\mathbb{R}}\)-invariant set
\(\tilde{\Omega}\subset\Omega\)
of full measure with sublinear growth
$$ \lim_{t\rightarrow\infty}\frac{\|W_{i}(t)\|}{t}=0 $$
of ℙ-measure one. In addition,
$$ \lim_{t\rightarrow\infty}\frac{|y_{i}(\theta_{t}\omega_{i})|}{|t|}=0 \quad\textit{and} \quad \lim _{t\rightarrow\infty}\frac{\int_{0}^{t}y_{i}(\theta_{s}\omega_{i})\,ds}{t}=0, $$
where
\(\omega\in\tilde{\Omega}\), \(i=1,2\).
By introducing the transformation \(\psi=\frac{dv}{dt}+\rho v\) with ρ a positive constant which satisfies \(\rho<\nu\), system (3.1) becomes
$$\begin{aligned}& i\frac{du}{dt}+\triangle u+i\alpha u+uv=f+\epsilon u\frac{dW_{1}}{dt},\quad x\in\mathbb{R}^{n}, \end{aligned}$$
(3.2)
$$\begin{aligned}& \frac{dv}{dt}=\psi-\rho v,\quad x\in\mathbb{R}^{n}, \end{aligned}$$
(3.3)
$$\begin{aligned}& \frac{d\psi}{dt}+(\nu-\rho)\psi+ \bigl[\mu-\rho(\nu-\rho)-\triangle \bigr]v- \beta |u|^{2}=g+\epsilon\delta\frac{dW_{2}}{dt},\quad x\in \mathbb{R}^{n}. \end{aligned}$$
(3.4)
In order to prove the existence of global solutions of (3.1), we introduce the processes
$$\tilde{u}(t)=z(t)u(t) $$
and
$$\tilde{\psi}(t)=\psi(t)-\epsilon\delta z_{2}(\theta_{t} \omega), $$
where \(z(t)=e^{i\epsilon z_{1}(\theta_{t}\omega)}\) satisfies the stochastic differential equation
$$ dz(t)=-\frac{\epsilon^{2}}{2}z(t)\,dt+i\epsilon z(t)\,dz_{1}. $$
(3.5)
Then system (3.2)-(3.4) can be changed into the following system:
$$ \left \{ \begin{array}{@{}l} i\frac{d\tilde{u}}{dt}+\triangle\tilde{u}+(i\alpha+\frac{\epsilon ^{2}}{2})\tilde{u}+\tilde{u}v-fz=0,\\ \frac{dv}{dt}+\rho v=\tilde{\psi}+\epsilon\delta z_{2},\\ \frac{d\tilde{\psi}}{dt}+(\nu-\rho)\tilde{\psi}+[\mu-\rho(\nu-\rho )-\triangle]v-\beta|\tilde{u}z^{-1}|^{2}+\epsilon(\nu-\rho)\delta z_{2}=g, \end{array} \right . $$
(3.6)
with the initial data \(\tilde{u}_{0}=u_{0}\), \(v_{0}=v_{0}\) and \(\tilde{\psi}_{0}=\psi_{0}-\epsilon\delta z_{2}(\omega)\).
Lemma 3.2
Let
\(f\in\mathbb{L}^{2}\). Then the solution of the first equation in (3.6) satisfies
$$ \|\tilde{u}\|^{2}\leq e^{-\alpha t}\|\tilde{u}_{0} \|^{2}+\frac{4}{\alpha^{2}}\|f\|^{2}. $$
Proof
Taking the imaginary part of the inner product of (3.6) with \(\tilde{u}\), we obtain
$$ \frac{d}{dt}\|\tilde{u}\|^{2}+2\alpha\|\tilde{u} \|^{2}=2\operatorname{Im}\int_{\mathbb {R}^{n}}(fz,\tilde{u})\,dx. $$
(3.7)
Obviously, the right-hand side of (3.7) is bounded by
$$ 2\operatorname{Im}\int_{\mathbb{R}^{n}}(fz,\tilde{u})\,dx\leq\alpha\|\tilde{u} \|^{2}+\frac {4}{\alpha}\|f\|^{2}\|z\|^{2}. $$
Thus we have
$$ \frac{d}{dt}\|\tilde{u}\|^{2}+\alpha\|\tilde{u}\|^{2} \leq\frac{4}{\alpha}\| f\|^{2}\|z\|^{2}\leq \frac{4}{\alpha}\|f\|^{2}. $$
By Gronwall’s lemma we get
$$ \|\tilde{u}\|^{2}\leq e^{-\alpha t}\|\tilde{u}_{0} \|^{2}+\frac{4}{\alpha^{2}}\|f\|^{2}. $$
The proof is completed. □
Remark 3.1
By Lemma 3.2, we know that there is \(T_{1}>0\) such that \(\|\tilde{u}\|\) is bounded for \(t>T_{1}\), i.e., \(\|\tilde{u}\| \leq M_{1}\), \(t>T_{1}\).
Lemma 3.3
Let
\(f\in\mathbb{L}^{4}\). Then, for any
\(m\geq0\), the solution of the first equation in (3.1) satisfies
$$ \int_{|x|\geq m}|\tilde{u}|^{4}\,dx\leq e^{-\alpha t} \int_{|x|\geq m}|\tilde{u}_{0}|^{4}\,dx+ \frac{64}{\alpha^{4}}\int_{|x|\geq m}|f|^{4}\,dx $$
for all
\(\omega\in\Omega\).
Proof
The proof is similar to the proof in Lemma 3.2, so we omit it here. □
Here and after, \(\mathbb{I}\) denotes the space \(\mathbb{L}^{2}(\mathbb{R}^{n})\times \mathbb{H}^{1}(\mathbb{R}^{n})\times\mathbb{L}^{2}(\mathbb{R}^{n})\). By Galerkin’s method, it is easy to prove that, for ℙ-a.e. \(\omega\in\Omega\) and for all \((\tilde{u}_{0},v_{0},\tilde{\psi}_{0})\in\mathbb{I}\), system (3.6) has a unique solution \((\tilde{u}(\cdot,\omega,\tilde{u}_{0}),v(\cdot,\omega,v_{0}),\tilde{\psi }(\cdot,\omega,\tilde{\psi}_{0})) \in\mathbb{C}([0,\infty),\mathbb{I})\) with \(\tilde{u}(0,\omega,\tilde{u}_{0})=\tilde{u}_{0}\), \(v(0,\omega,v_{0})=v_{0}\) and \(\tilde{\psi}(0,\omega,\tilde{\psi}_{0})=\tilde{\psi}_{0}\). Furthermore, the solution is continuous with respect to \((\tilde{u}_{0},v_{0},\tilde{\psi}_{0})\in\mathbb{I}\). To indicate the dependence of \((u,v,\psi)\) on the initial data \((u_{0},v_{0},\psi_{0})\), we define a mapping
$$ \phi_{\epsilon}:\mathbb{R}^{+}\times\Omega\times\mathbb{I} \rightarrow \mathbb{I} $$
by
$$\begin{aligned}[b] \phi_{\epsilon} \bigl(t,\omega,(u_{0},v_{0}, \psi_{0}) \bigr)&= \bigl(u(t,\omega,u_{0}),v(t,\omega ,v_{0}),\psi(t,\omega,\psi_{0}) \bigr) \\ &= \bigl(\tilde{u}(t,\omega,\tilde{u}_{0})z^{-1}(t),v(t, \omega,v_{0}),\tilde{\psi }(t,\omega,\tilde{\psi}_{0})+ \epsilon\delta z_{2}(\theta_{t}\omega) \bigr) \end{aligned} $$
for all \((t,\omega,(u_{0},v_{0},\psi_{0}))\in\mathbb{R}^{+}\times\Omega\times\mathbb{I}\).
It is obvious that \(\phi_{\epsilon}\) satisfies all conditions in Definition 2.2. Therefore, \(\phi_{\epsilon}\) is a continuous random dynamical system associated with (3.1). It is easy to verify that \(\phi_{\epsilon}\) satisfies
$$ \phi_{\epsilon} \bigl(t,\theta_{-t}\omega,(u_{0},v_{0}, \psi_{0}) \bigr)= \bigl(u(t,\theta_{-t}\omega,u_{0}),v(t, \theta_{-t}\omega,v_{0}),\psi(t,\theta _{-t} \omega, \psi_{0}) \bigr) $$
for ℙ-a.e. \(\omega\in\Omega\) and \(t\geq0\).
The following lemma shows that \(\phi_{\epsilon}\) has a closed bounded random absorbing set in \(\mathbb{D}\), which verifies the first condition (I) in Proposition 2.8.
Lemma 3.4
Let
\(f,g\in\mathbb{L}^{2}\), \(\delta\in\mathbb{H}^{1}\). Assume that
\(\mu-\rho(\nu-\rho)>0\). Then there exists
\(\{\mathbb{K}(\omega)\}_{\omega\in\Omega}\in\mathbb{D}\)
such that
\(\{\mathbb{K}(\omega)\}_{\omega\in\Omega}\)
is a random absorbing set for
ϕ
in
\(\mathbb{D}\), that is, for any
\(\mathbb{B}=\{\mathbb{B}(\omega)\}_{\omega\in\Omega}\in\mathbb{D}\)
and ℙ-a.e. \(\omega\in\Omega\), there is
\(T_{\mathbb{B}}(\omega)>0\)
such that
$$ \phi_{\epsilon} \bigl(t,\theta_{-t}\omega,\mathbb{B}( \theta_{-t}\omega ) \bigr)\subseteq\mathbb{K}(\omega)\quad \textit{for all } t\geq T_{\mathbb{B}}(\omega). $$
Proof
Taking the imaginary part of the inner product of the first equation of (3.6) with \(\tilde{u}\), we have
$$ \frac{d}{dt}\|\tilde{u}\|^{2}+2\alpha\|\tilde{u} \|^{2}=2\operatorname{Im}(fz,\tilde{u}). $$
(3.8)
Taking the inner product of the third equation of (3.6) with \(\tilde{\psi}\), we get
$$\begin{aligned} &\frac{d}{dt}\|\tilde{\psi}\|^{2}+2(\nu-\rho)\| \tilde{\psi}\|^{2}+2 \bigl[\mu -\rho(\nu-\rho) \bigr](v,\tilde{\psi}) -2( \triangle v,\tilde{\psi}) \\ &\quad{}-2\beta \bigl(\bigl|\tilde{u}z^{-1}\bigr|^{2},\tilde{\psi} \bigr)+2\epsilon(\nu-\rho) \bigl(\delta z_{2}(\theta_{t} \omega),\tilde{\psi} \bigr)=2(g,\tilde{\psi}). \end{aligned}$$
(3.9)
Note that
$$ 2(v,\tilde{\psi})=2 \biggl(v,\frac{dv}{dt}+\rho v-\epsilon \delta z_{2}(\theta_{t}\omega) \biggr)=\frac{d}{dt}\|v \|^{2}+2\rho\|v\|^{2}-2\epsilon \bigl(v,\delta z_{2}( \theta_{t}\omega) \bigr) $$
(3.10)
and
$$\begin{aligned} -2(\triangle v,\tilde{\psi}) =&\frac{d}{dt}\|\nabla v \|^{2}+2\rho\|\nabla v\|^{2}+2\epsilon \bigl(\nabla v, z_{2}(\theta_{t}\omega)\nabla\delta \bigr) \\ &{}+2\epsilon \bigl(\nabla v,\delta\nabla z_{2}(\theta_{t} \omega) \bigr). \end{aligned}$$
(3.11)
Summing up (3.8)-(3.11), we have
$$\begin{aligned} &\frac{d}{dt} \bigl(\|\tilde{u}\|^{2}+ \bigl[\mu- \rho(\nu-\rho) \bigr]\|v\|^{2}+\|\nabla v\|^{2} +\|\tilde{ \psi} \|^{2} \bigr)+2\alpha\|\tilde{u}\|^{2} \\ &\qquad{}+2\rho \bigl[\mu-\rho(\nu-\rho) \bigr]\|v\|^{2}+2\rho\|\nabla v \|^{2}+2(\nu-\rho)\|\tilde{\psi}\|^{2} \\ &\quad=2\operatorname{Im}(fz,\tilde{u})-2\epsilon \bigl(\nabla v, z_{2}( \theta_{t}\omega)\nabla\delta \bigr) -2\epsilon \bigl(\nabla v, \delta \nabla z_{2}(\theta_{t}\omega) \bigr) \\ &\qquad{}+2\epsilon \bigl[\mu-\rho(\nu-\rho) \bigr] \bigl(v,\delta z_{2}( \theta_{t}\omega) \bigr)+2\beta \bigl(\bigl| \tilde{u}z^{-1}\bigr|^{2}, \tilde{\psi} \bigr) \\ &\qquad{}-2\epsilon(\nu-\rho) \bigl(\delta z_{2}(\theta_{t} \omega),\tilde{\psi} \bigr)+2(g,\tilde{\psi}). \end{aligned}$$
(3.12)
Now, we estimate each term on the right-hand side of (3.12). By Young’s inequality, we have
$$\begin{aligned}& 2\bigl|\operatorname{Im}(fz,\tilde{u})\bigr|\leq\alpha\|\tilde{u}\|^{2}+\frac{4}{\alpha}\|f\| ^{2}, \\& 2\bigl|\epsilon \bigl(\nabla v, z_{2}(\theta_{t}\omega)\nabla \delta \bigr)\bigr|\leq\frac{\rho }{2}\|\nabla v\|^{2}+\frac{8\epsilon^{2}}{\rho} \| \nabla\delta\|^{2}\bigl\| z_{2}(\theta_{t}\omega) \bigr\| ^{2}, \\& 2\bigl|\epsilon \bigl(\nabla v, \delta\nabla z_{2}(\theta_{t} \omega) \bigr)\bigr|\leq\frac{\rho }{2}\|\nabla v\|^{2}+\frac{8\epsilon^{2}}{\rho} \|\delta\|^{2}\bigl\| \nabla z_{2}(\theta_{t}\omega) \bigr\| ^{2}, \\& \begin{aligned}[b] &2\bigl|\epsilon \bigl[\mu-\rho(\nu-\rho) \bigr] \bigl(v,\delta z_{2}(\theta_{t}\omega) \bigr)\bigr|\\ &\quad\leq\rho \bigl(\mu-\rho( \nu-\rho) \bigr)\|v\|^{2} +\frac{4\epsilon^{2}}{\rho(\mu-\rho(\nu-\rho))}\|\delta\|^{2}\bigl\| z_{2}(\theta _{t}\omega)\bigr\| ^{2}, \end{aligned} \\& 2\beta\bigl| \bigl(\bigl|\tilde{u}z^{-1}\bigr|^{2},\tilde{\psi} \bigr)\bigr|\leq \frac{\nu-\rho}{3}\| \tilde{\psi}\|^{2}+\frac{12\beta^{2}}{\nu-\rho}\|\tilde{u} \|^{4}, \\& 2\bigl|\epsilon(\nu-\rho) \bigl(\delta z_{2}(\theta_{t}\omega), \tilde{\psi} \bigr)\bigr|\leq \frac{\nu-\rho}{3}\|\tilde{\psi}\|^{2}+12 \epsilon^{2}(\nu-\rho)\|\delta\| ^{2}\bigl\| z_{2}( \theta_{t}\omega)\bigr\| ^{2}, \\& 2\bigl|(g,\tilde{\psi})\bigr|\leq\frac{\nu-\rho}{3}\|\tilde{\psi}\|^{2}+ \frac {12}{\nu-\rho}\|g\|^{2}. \end{aligned}$$
By combining the above estimates with (3.12), we obtain
$$\begin{aligned} &\frac{d}{dt} \bigl(\|\tilde{u}\|^{2}+ \bigl[\mu- \rho(\rho-\mu) \bigr]\|v\|^{2}+\|\nabla v\|^{2} +\|\tilde{ \psi} \|^{2} \bigr)+\alpha\|\tilde{u}\|^{2} \\ &\qquad{}+\rho \bigl[\mu-\rho(\rho-\mu) \bigr]\|v\|^{2}+\rho\|\nabla v \|^{2}+(\nu-\rho)\|\tilde{\psi}\|^{2} \\ &\quad\leq\frac{4}{\alpha}\|f\|^{2}+\frac{8\epsilon^{2}}{\rho}\|\delta \|^{2}\bigl\| \nabla z_{2}(\theta_{t}\omega) \bigr\| ^{2}+\frac{12\beta^{2}}{\nu-\rho}\|\tilde{u}\|^{4}+\frac {12}{\nu-\rho} \|g\|^{2} \\ &\qquad{}+ \biggl[\frac{8\epsilon^{2}}{\rho}\|\nabla\delta\|^{2}+ \frac{4\epsilon^{2}}{\rho (\mu-\rho(\nu-\rho))}\|\delta\|^{2}+12\epsilon^{2}(\nu-\rho)\| \delta\|^{2} \biggr]\bigl\| z_{2}(\theta_{t}\omega) \bigr\| ^{2}. \end{aligned}$$
(3.13)
Furthermore, by Lemma 3.2, for \(t\geq0\)
$$ \|\tilde{u}\|^{4}\leq e^{-2\alpha t}\| \tilde{u}_{0}\|^{4}+\frac{8}{\alpha^{2}}e^{-\alpha t} \|u_{0}\|^{2}\|f\|^{2}+\frac{16}{\alpha^{4}}\|f \|^{4}. $$
(3.14)
Therefore, by (3.13) and (3.14), we have
$$\begin{aligned} &\frac{d}{dt} \bigl(\|\tilde{u}\|^{2}+ \bigl[\mu- \rho(\rho-\mu) \bigr]\|v\|^{2}+\|\nabla v\|^{2} +\|\tilde{ \psi} \|^{2} \bigr)+\alpha\|\tilde{u}\|^{2} \\ &\qquad{}+\rho \bigl[\mu-\rho(\rho-\mu) \bigr]\|v\|^{2}+\rho\|\nabla v \|^{2}+(\nu-\rho)\|\tilde{\psi}\|^{2} \\ &\quad\leq\frac{4}{\alpha}\|f\|^{2}+\frac{8\epsilon^{2}}{\rho}\|\delta \|^{2}\bigl\| \nabla z_{2}(\theta_{t}\omega) \bigr\| ^{2}+\frac{12\beta^{2}}{\nu-\rho} \biggl[e^{-2\alpha t}\|\tilde{u}_{0} \|^{4} \\ &\qquad{}+\frac{8}{\alpha^{2}}e^{-\alpha t}\|u_{0}\|^{2} \|f\|^{2}+\frac{16}{\alpha^{4}}\|f\|^{4} \biggr]+ \frac{12}{\nu-\rho}\|g\|^{2} \\ &\qquad{}+ \biggl[\frac{8\epsilon^{2}}{\rho}\|\nabla\delta\|^{2}+ \frac{4\epsilon^{2}}{\rho (\mu-\rho(\nu-\rho))}\|\delta\|^{2}+12\epsilon^{2}(\nu-\rho)\| \delta\|^{2} \biggr]\bigl\| z_{2}(\theta_{t}\omega) \bigr\| ^{2}. \end{aligned}$$
(3.15)
Let \(C_{1}:=\min\{\alpha,\rho,(\nu-\rho)\}>0\). Then, by (3.15), we have
$$ \frac{d}{dt}E_{1}(\tilde{u},v,\tilde{ \psi})+C_{1}E_{1}(\tilde{u},v,\tilde{\psi })\leq F \bigl(f,g,z_{2}(\theta_{t}\omega),\nabla z_{2}( \theta_{t}\omega) \bigr), $$
(3.16)
where
$$ E_{1}(\tilde{u},v,\tilde{\psi})=\|\tilde{u}\|^{2}+ \bigl[ \mu- \rho(\rho-\mu) \bigr]\|v\| ^{2}+\|\nabla v\|^{2} +\| \tilde{ \psi}\|^{2}, $$
and
$$\begin{aligned} &F \bigl(f,g,z_{2}(\theta_{t}\omega),\nabla z_{2}( \theta_{t}\omega) \bigr) \\ &\quad=\frac{4}{\alpha}\|f\|^{2}+\frac{8\epsilon^{2}}{\rho}\|\delta \|^{2}\bigl\| \nabla z_{2}(\theta_{t}\omega) \bigr\| ^{2}+\frac{12}{\nu-\rho}\|g\|^{2} \\ &\qquad{}+\frac{12\beta^{2}}{\nu-\rho} \biggl[e^{-2\alpha t}\|\tilde{u}_{0} \|^{4}+\frac{8}{\alpha^{2}}e^{-\alpha t}\|u_{0} \|^{2}\|f\|^{2}+\frac{16}{\alpha^{4}}\|f\|^{4} \biggr] \\ &\qquad{}+ \biggl[\frac{8\epsilon^{2}}{\rho}\|\nabla\delta\|^{2}+ \frac{4\epsilon^{2}}{\rho (\mu-\rho(\nu-\rho))}\|\delta\|^{2}+12\epsilon^{2}(\nu-\rho)\| \delta\|^{2} \biggr]\bigl\| z_{2}(\theta_{t}\omega) \bigr\| ^{2}. \end{aligned}$$
Note that \(z_{1}(\theta_{t}\omega)=y_{1}(\theta_{t}\omega_{1})\) and \(z_{2}(\theta_{t}\omega)=\delta y_{2}(\theta_{t}\omega_{2})\), \(\delta\in\mathbb{H}^{1}(\mathbb{R}^{n})\), therefore, by Lemma 3.1, F is bounded by
$$ C\sum^{2}_{i=1} \bigl(\bigl|y_{i}(\theta_{t}\omega_{i})\bigr|^{2}+\bigl|y_{i}( \theta_{t}\omega _{i})\bigr|^{p} \bigr)+C=P_{1}( \theta_{t}\omega)+C. $$
(3.17)
By Proposition 4.3.3 in [5], there exists a tempered function \(r(\omega)>0\) such that \(r(\theta_{t}\omega)\leq e^{\frac{C}{2}|t|}r(\omega)\). Therefore, by (3.17), we find that for ℙ-a.e. \(\omega\in\Omega\),
$$ P_{1}(\theta_{s}\omega)\leq Ce^{\frac{C}{2}|s|}r(\omega), \quad \forall s\in\mathbb{R}. $$
By replacing ω by \(\theta_{-t}\omega\), we get from (3.16) and (3.17)
$$\begin{aligned} &E_{1} \bigl(\tilde{u} \bigl(t,\theta_{-t} \omega,\tilde{u}_{0}(\theta_{-t}\omega ) \bigr),v \bigl(t, \theta_{-t}\omega,v_{0}(\theta_{-t}\omega) \bigr), \tilde{\psi} \bigl(t,\theta_{-t}\omega,\tilde{\psi}_{0}( \theta_{-t}\omega ) \bigr) \bigr) \\ &\quad\leq e^{-C_{1}t}E_{1} \bigl(\tilde{u}_{0}( \theta_{-t}\omega),v_{0}(\theta_{-t}\omega), \tilde{\psi}_{0}(\theta_{-t}\omega) \bigr)+\epsilon\int _{0}^{t}e^{C_{1}(s-t)}P_{1}( \theta_{s-t}\omega)\,ds+C_{2} \\ &\quad\leq e^{-C_{1}t}E_{1} \bigl(\tilde{u}_{0}( \theta_{-t}\omega),v_{0}(\theta_{-t}\omega), \tilde{\psi}_{0}(\theta_{-t}\omega) \bigr)+\epsilon\int _{-t}^{0}e^{C_{1}s}P_{1}( \theta_{s}\omega)\,ds+C_{2} \\ &\quad\leq e^{-C_{1}t}E_{1} \bigl(\tilde{u}_{0}( \theta_{-t}\omega),v_{0}(\theta_{-t}\omega), \tilde{\psi}_{0}(\theta_{-t}\omega) \bigr)+\epsilon C_{1}\int_{-t}^{0}e^{\frac {C_{1}}{2}s}r( \omega)\,ds+C_{2} \\ &\quad\leq e^{-C_{1}t}E_{1} \bigl(\tilde{u}_{0}( \theta_{-t}\omega),v_{0}(\theta_{-t}\omega ), \tilde{\psi}_{0}(\theta_{-t}\omega) \bigr)+\epsilon C_{1}r(\omega)+C_{2}. \end{aligned}$$
(3.18)
Note that
$$\begin{aligned} &\phi \bigl(t,\theta_{-t}\omega,(u_{0},v_{0}, \psi_{0}) \bigr) \\ &\quad= \bigl(\tilde{u} \bigl(t,\theta_{-t}\omega,u_{0}e^{-i\epsilon z_{1}(\omega)} \bigr)z^{-1},v(t,\theta_{-t}\omega,v_{0}), \tilde{ \psi} \bigl(t,\theta_{-t}\omega,\tilde{\psi}_{0}+\epsilon \delta z_{2}(\omega) \bigr)+\epsilon\delta z_{2}( \theta_{-t}\omega) \bigr). \end{aligned}$$
Therefore, by (3.18) we know that there exists a positive constant \(C_{3}\) such that, for all \(t\geq0\),
$$\begin{aligned} &\bigl\| \phi \bigl(t,\theta_{-t}\omega,(u_{0},v_{0}, \psi_{0}) (\theta_{-t}\omega) \bigr)\bigr\| ^{2}_{\mathbb{I}} \\ &\quad=\bigl\| \tilde{u} \bigl(t,\theta_{-t}\omega,u_{0}( \theta_{-t}\omega)e^{-i\epsilon z_{1}(\theta_{-t}\omega)} \bigr)z^{-1} \bigr\| ^{2} \\ &\qquad{}+\bigl\| v \bigl(t,\theta_{-t}\omega,v_{0}( \theta_{-t}\omega) \bigr)\bigr\| ^{2}_{\mathbb{H}^{1}} +\bigl\| \tilde{\psi} \bigl(t,\theta_{-t}\omega,\psi_{0}(\theta_{-t} \omega)+\epsilon \delta z_{2}(\omega) \bigr)+\epsilon\delta z_{2}(\omega)\bigr\| ^{2} \\ &\quad\leq\bigl\| \tilde{u} \bigl(t,\theta_{-t}\omega,u_{0}( \theta_{-t}\omega )e^{-i\epsilon z_{1}(\theta_{-t}\omega)} \bigr)\bigr\| ^{2}+\bigl\| v \bigl(t, \theta_{-t}\omega,v_{0}(\theta _{-t}\omega) \bigr) \bigr\| ^{2}_{\mathbb{H}^{1}} \\ &\qquad{}+2\bigl\| \tilde{\psi} \bigl(t,\theta_{-t}\omega, \psi_{0}(\theta_{-t}\omega )+\epsilon\delta z_{2}( \theta_{-t}\omega) \bigr)\bigr\| ^{2}+2\epsilon^{2}\| \delta \|^{2}\bigl\| z_{2}(\omega)\bigr\| ^{2} \\ &\quad\leq C_{3}e^{-C_{1}t}E_{1} \bigl(u_{0}( \theta_{-t}\omega),v_{0}(\theta_{-t}\omega),\psi _{0}(\theta_{-t}\omega) \bigr) +C_{3}e^{-C_{1}t} \epsilon^{2}\|\delta\|^{2}\bigl\| z_{2}( \theta_{-t}\omega)\bigr\| ^{2} \\ &\qquad{}+2\epsilon^{2}\|\delta\|^{2}\bigl\| z_{2}( \omega)\bigr\| ^{2}+\epsilon C_{1}r(\omega)+C_{2}. \end{aligned}$$
(3.19)
By definition of \(z_{2}(\omega)\), it is easy to see that \(\|z_{2}(\omega)\|^{2}\) is tempered. In addition, \(\{\mathbb{B}(\omega)\}_{\omega\in\Omega}\subset\mathbb{D}\) is also tempered by assumption. Therefore, for \((u_{0}(\theta_{-t}\omega),v_{0}(\theta_{-t}\omega),\psi_{0}(\theta_{-t}\omega ))\in\mathbb{B}(\theta_{-t}\omega)\), there is \(T_{\mathbb{B}}(\omega)>0\) such that for all \(t\geq T_{\mathbb{B}}(\omega)\),
$$\begin{aligned} &e^{-C_{1}t}E_{1} \bigl(u_{0}(\theta_{-t} \omega),v_{0}(\theta_{-t}\omega),\psi _{0}( \theta_{-t}\omega) \bigr) +C_{3}e^{-C_{1}t} \epsilon^{2}\|\delta\|^{2}\bigl\| z_{2}( \theta_{-t}\omega)\bigr\| ^{2} \leq\epsilon C_{4}r(\omega)+C_{4}, \end{aligned}$$
which along with (3.19) leads to
$$ \bigl\| \phi \bigl(t,\theta_{-t}\omega,(u_{0},v_{0}, \psi_{0}) (\theta_{-t}\omega) \bigr)\bigr\| ^{2}_{\mathbb{I}} \leq C_{5}+\epsilon C_{5}r(\omega)+2\epsilon^{2} \| \delta\|^{2}\bigl\| z_{2}(\omega)\bigr\| ^{2}. $$
Given \(\omega\in\Omega\), define
$$\begin{aligned} \mathbb{K}(\omega) = \bigl\{ (u,v,\psi)\in\mathbb{I}:\|u\|^{2}+\|v \|^{2}_{\mathbb{H}^{1}}+\|\psi\|^{2} \leq C+\epsilon Cr( \omega)+2\epsilon^{2}\|\delta\|^{2}\bigl\| z_{2}(\omega) \bigr\| ^{2} \bigr\} . \end{aligned}$$
Then \(\{\mathbb{K}(\omega)\}_{\omega\in\Omega}\) is an absorbing set for ϕ in \(\mathbb{D}\), which completes the proof. □
Lemma 3.5
Let
\(f,g\in\mathbb{H}^{1}\), \(\delta\in\mathbb{H}^{2}\), \(\mathbb{B}=\{\mathbb{B}(\omega)\}_{\omega\in\Omega}\in\mathbb{D}\)
and
\((u_{0}(\omega),v_{0}(\omega),\psi_{0}(\omega))\in\mathbb{B}(\omega)\). Assume that
\(\alpha(\nu-\rho)>12M_{1}\beta^{2}\), \(\alpha(\mu-\rho(\nu-\rho))>8M_{1}\). Then, for ℙ-a.e. \(\omega\in\Omega\), there exists
\(T_{\mathbb{B}}'(\omega)>0\)
such that for all
\(t\geq T_{\mathbb{B}}'\),
$$\begin{aligned} &\bigl\| u \bigl(t,\theta_{-t}\omega,u_{0}(\theta_{-t} \omega) \bigr)\bigr\| ^{2}_{\mathbb{H}^{1}}+\bigl\| v \bigl(t,\theta_{-t} \omega,v_{0}(\theta_{-t}\omega) \bigr)\bigr\| ^{2}_{\mathbb{H}^{2}} +\bigl\| \psi \bigl(t,\theta_{-t}\omega,\psi_{0}( \theta_{-t}\omega) \bigr)\bigr\| ^{2}_{\mathbb {H}^{1}}\leq R(\omega), \end{aligned}$$
where
\(R(\omega)\)
is a positive tempered random function.
Proof
Taking the real part of the inner product of the first equation in system (3.6) with \(-\triangle\tilde{u}\) in \(\mathbb{L}^{2}(\mathbb{R}^{n})\), we have that
$$ \frac{d}{dt}\|\nabla\tilde{u}\|^{2}+2\alpha\| \nabla\tilde{u}\|^{2}+2(\nabla \tilde{u},\tilde{u}\nabla v) +2\operatorname{Im} \bigl( \nabla\tilde{u},\nabla(fz) \bigr)=0. $$
(3.20)
Taking the inner product of the third equation in system (3.6) with \(-\triangle\tilde{\psi}\) in \(\mathbb{L}^{2}(\mathbb{R}^{n})\), we find
$$ \begin{aligned}[b] &\frac{d}{dt}\|\nabla\tilde{\psi}\|^{2}+2(\nu- \rho)\|\nabla\tilde{\psi}\| ^{2}+2(\triangle\tilde{\psi},\triangle v)-2 \bigl[\mu-\rho(\nu-\rho) \bigr](\triangle\tilde{\psi},v) \\ &\quad{}+2\beta \bigl(\triangle\tilde{\psi},\bigl|\tilde{u}z^{-1}\bigr|^{2} \bigr)-2\epsilon(\nu-\rho ) \bigl(\triangle\tilde{\psi},\delta z_{2}( \theta_{-t}\omega) \bigr)+2(\triangle\tilde{\psi},g)=0. \end{aligned} $$
(3.21)
By the second equation in system (3.6), we have
$$\begin{aligned} &2(\triangle\tilde{\psi},\triangle v)-2 \bigl[\mu-\rho(\nu-\rho) \bigr](\triangle\tilde{\psi},v) \\ &\quad=2 \bigl(\triangle v_{t}+\rho\triangle v-\epsilon\triangle \bigl(z_{2}(\theta_{t}\omega)\delta \bigr), \triangle v \bigr) \\ &\qquad{}-2 \bigl[\mu-\rho(\nu-\rho) \bigr] \bigl(\triangle v_{t}+ \rho \triangle v-\epsilon\triangle \bigl(z_{2}(\theta_{t} \omega) \delta \bigr),v \bigr) \\ &\quad=\frac{d}{dt} \bigl(\|\triangle v\|^{2}+ \bigl(\mu-\rho( \nu- \rho) \bigr)\|\nabla v\|^{2} \bigr)+2 \bigl(\rho\|\triangle v \|^{2}+ \bigl(\mu-\rho(\nu-\rho) \bigr)\|\nabla v\|^{2} \bigr) \\ &\qquad{}-2\epsilon \bigl(z_{2}(\theta_{t}\omega)\triangle \delta,\triangle v \bigr) -2\epsilon \bigl(\delta\triangle z_{2}( \theta_{t}\omega),\triangle v \bigr) \\ &\qquad{}-2\epsilon \bigl[\mu-\rho(\nu-\rho) \bigr] \bigl(z_{2}( \theta_{t}\omega)\nabla\delta,\nabla v \bigr) \\ &\qquad{}-2\epsilon \bigl[\mu-\rho(\nu-\rho) \bigr] \bigl(\delta\nabla z_{2}(\theta_{t}\omega),\nabla v \bigr). \end{aligned}$$
(3.22)
Then it follows from (3.20)-(3.22) that
$$\begin{aligned} &\frac{d}{dt} \bigl(\|\nabla\tilde{u}\|^{2}+\| \nabla\tilde{\psi}\|^{2}+\| \triangle v\|^{2}+ \bigl(\mu-\rho( \nu-\rho) \bigr)\|\nabla v\|^{2} \bigr) \\ &\qquad{}+2\alpha\|\nabla\tilde{u}\|^{2}+2(\nu-\rho)\|\nabla\tilde{ \psi}\| ^{2}+2\rho\|\triangle v\|^{2}+2 \bigl(\mu-\rho(\nu- \rho) \bigr)\|\nabla v\|^{2} \\ &\quad=-2\operatorname{Im} \bigl(\nabla\tilde{u},\nabla(fz) \bigr)-2(\nabla\tilde{u},\tilde{u} \nabla v)-2\beta \bigl(\triangle\tilde{\psi},\bigl|\tilde{u}z^{-1}\bigr|^{2} \bigr) \\ &\qquad{}+2\epsilon(\nu-\rho) \bigl(\triangle\tilde{\psi},z_{2}( \theta_{t}\omega)\delta \bigr) -2(\triangle\tilde{\psi},g)+2\epsilon \bigl(z_{2}(\theta_{t}\omega)\triangle\delta ,\triangle v \bigr) \\ &\qquad{}+2\epsilon \bigl(\delta\triangle z_{2}(\theta_{t} \omega),\triangle v \bigr) +2\epsilon \bigl[\mu-\rho(\nu-\rho) \bigr] \bigl(z_{2}(\theta_{t}\omega)\nabla\delta,\nabla v \bigr) \\ &\qquad{}+2\epsilon \bigl[\mu-\rho(\nu-\rho) \bigr] \bigl(\delta\nabla z_{2}(\theta_{t}\omega),\nabla v \bigr). \end{aligned}$$
(3.23)
Now, we estimate each term on the right-hand side of (3.23). By Young’s inequality, we get
$$\begin{aligned}& \begin{aligned}[b] \bigl|-2\operatorname{Im} \bigl(\nabla\tilde{u},\nabla(fz) \bigr)\bigr| &=2\bigl|\operatorname{Im}(\nabla\tilde{u},z\nabla f)+\operatorname{Im}( \nabla\tilde{u},f\nabla z)\bigr|\\ &\leq\frac{\alpha}{2}\|\nabla\tilde{u} \|^{2} +\frac{8}{\alpha} \bigl(\|\nabla f\|^{2}+ \epsilon^{2}\|f \|^{2}\|\nabla z_{1}\|^{2} \bigr), \end{aligned} \\& \bigl|-2(\nabla\tilde{u},\tilde{u}\nabla v)\bigr|\leq\frac{\alpha}{2}\|\nabla \tilde{u} \|^{2}+ \frac{8}{\alpha}\|\tilde{u}\|^{2}\|\nabla v \|^{2}, \\& \bigl|-2\beta \bigl(\triangle\tilde{\psi},\bigl|\tilde{u}z^{-1}\bigr|^{2} \bigr)\bigr| \leq\frac{\nu-\rho}{3}\|\nabla\tilde{\psi}\|^{2}+ \frac{12\beta^{2}}{\nu-\rho }\|\tilde{u}\|^{2}\|\nabla\tilde{u}\|^{2}, \\& \begin{aligned}[b] &\bigl|2\epsilon(\nu-\rho) \bigl(\triangle\tilde{\psi},z_{2}( \theta_{t}\omega)\delta \bigr)\bigr|\\ &\quad\leq\frac{\nu-\rho}{3}\|\nabla\tilde{ \psi}\|^{2} +12\epsilon^{2}(\nu-\rho) \bigl(\|\delta \|^{2}\bigl\| \nabla z_{2}(\theta_{t}\omega)\bigr\| ^{2}+\bigl\| z_{2}(\theta_{t}\omega)\bigr\| ^{2}\| \nabla\delta\|^{2} \bigr), \end{aligned} \\& \bigl|-2(\triangle\tilde{\psi},g)\bigr|\leq\frac{\nu-\rho}{3}\|\nabla\tilde{\psi } \|^{2}+\frac{12}{\nu-\rho}\|\nabla g\|^{2}, \\& \bigl|2\epsilon \bigl(z_{2}(\theta_{t}\omega)\triangle\delta, \triangle v \bigr)\bigr| \leq\frac{\rho}{2}\|\triangle v\|^{2}+ \frac{8\epsilon^{2}}{\rho}\|\triangle \delta\|^{2}\bigl\| z_{2}( \theta_{t}\omega)\bigr\| ^{2}, \\& \bigl|2\epsilon \bigl(\delta\triangle z_{2}(\theta_{t}\omega), \triangle v \bigr)\bigr|\leq\frac {\rho}{2}\|\triangle v\|^{2}+ \frac{8\epsilon^{2}}{\rho}\|\delta\|^{2}\bigl\| \triangle z_{2}( \theta_{t}\omega)\bigr\| ^{2}, \\& \begin{aligned}[b] &\bigl|2\epsilon \bigl[\mu-\rho(\nu-\rho) \bigr] \bigl(z_{2}(\theta_{t}\omega)\nabla\delta,\nabla v \bigr)\bigr|\\ &\quad\leq\frac{\mu-\rho(\nu-\rho)}{2}\|\nabla v\|^{2} +8\epsilon^{2} \bigl[ \mu-\rho(\nu-\rho) \bigr]\bigl\| z_{2}(\theta_{t}\omega) \bigr\| ^{2}\|\nabla \delta\|^{2}, \end{aligned}\\& \begin{aligned}[b] &\bigl|2\epsilon \bigl[\mu-\rho(\nu-\rho) \bigr] \bigl(\delta\nabla z_{2}( \theta_{t}\omega),\nabla v \bigr)\bigr|\\ &\quad\leq\frac{\mu-\rho(\nu-\rho)}{2}\|\nabla v \|^{2} +8\epsilon^{2} \bigl[\mu-\rho(\nu-\rho) \bigr]\bigl\| \nabla z_{2}(\theta_{t}\omega)\bigr\| ^{2}\|\delta \|^{2}, \end{aligned} \end{aligned}$$
which along with (3.23), Remark 3.1 and assumption gives that there exists a positive constant
$$C_{6}=\min \biggl\{ \alpha-\frac{12M_{1}\beta^{2}}{\nu-\rho},\nu-\rho,\rho,\mu-\rho ( \nu-\rho)-\frac{8M_{1}}{\alpha} \biggr\} >0 $$
such that for \(t>T_{1}\)
$$ \frac{d}{dt}E_{2}(\tilde{u},v,\tilde{ \psi})+C_{6}E_{2}(\tilde{u},v,\tilde{\psi })\leq F \bigl(z_{1}(\theta_{t}\omega),z_{2}( \theta_{t}\omega) \bigr), $$
(3.24)
where
$$ E_{2}(\tilde{u},v,\tilde{\psi})=\|\nabla\tilde{u}\|^{2}+\| \nabla\tilde{\psi }\|^{2}+\|\triangle v\|^{2}+ \bigl(\mu-\rho( \nu-\rho) \bigr)\|\nabla v\|^{2}, $$
and
$$\begin{aligned} &F \bigl(z_{1}(\theta_{t}\omega),z_{2}( \theta_{t}\omega) \bigr) \\ &\quad=\frac{8}{\alpha} \bigl(\|\nabla f\|^{2}+\epsilon^{2} \|f\|^{2}\bigl\| \nabla z_{1}(\theta _{t}\omega) \bigr\| ^{2} \bigr) \\ &\qquad{}+24\epsilon^{2}(\nu-\rho) \bigl(\|\delta\|^{2}\bigl\| \nabla z_{2}(\theta_{t}\omega)\bigr\| ^{2}+ \bigl\| z_{2}(\theta_{t}\omega)\bigr\| ^{2}\|\nabla\delta \|^{2} \bigr)+\frac{12}{\nu-\rho}\| \nabla g\|^{2} \\ &\qquad{}+\frac{12\epsilon^{2}}{\rho}\bigl(\|\triangle\delta\|^{2}\bigl\| z_{2}( \theta_{t}\omega )\bigr\| ^{2}+\|\delta\|^{2}\bigl\| \triangle z_{2}(\theta_{t}\omega)\bigr\| ^{2}\bigr) \\ &\qquad{}+8\epsilon^{2} \bigl[\mu-\rho(\nu-\rho) \bigr] \bigl( \bigl\| z_{2}(\theta_{t}\omega)\bigr\| ^{2}\|\nabla \delta \|^{2}+\bigl\| \nabla z_{2}(\theta_{t}\omega) \bigr\| ^{2}\|\delta\|^{2} \bigr). \end{aligned}$$
Let
$$ P_{2}(\theta_{t}\omega)=C \bigl(\|\nabla z_{1} \|^{4}+\|z_{2}\|^{2}+\|\nabla z_{2} \|^{4}+\|\triangle z_{2}\|^{4} \bigr). $$
Since \(z_{1}(\theta_{t}\omega)=y_{1}(\theta_{t}\omega_{1})\) and \(z_{2}(\theta_{t}\omega)=\delta y_{2}(\theta_{t}\omega_{2})\), \(\delta\in\mathbb{H}^{2}(\mathbb{R}^{n})\). Therefore, by Lemma 3.1, we have
$$ P_{2}(\theta_{t}\omega)\leq C\sum _{i=1}^{2} \bigl(\bigl|y_{i}( \theta_{t}\omega_{i})\bigr|^{2}+\bigl|y_{i}( \theta_{t}\omega_{i})\bigr|^{p}_{p} \bigr)+C, $$
and
$$ P_{2}(\theta_{t}\omega)\leq Ce^{\frac{C}{2}|t|}r(\omega)+C \quad\mbox{for all }t\in\mathbb{R}. $$
By replacing ω by \(\theta_{-t}\omega\), it follows from Gronwall’s lemma that
$$\begin{aligned} &E_{2} \bigl(\tilde{u} \bigl(t,\theta_{-t}\omega, \tilde{u}_{0}(\theta_{-t}\omega) \bigr), v \bigl(t, \theta_{-t}\omega,v_{0}(\theta_{-t}\omega) \bigr), \tilde{\psi} \bigl(t,\theta _{-t}\omega,\tilde{\psi}_{0}( \theta_{-t}\omega) \bigr) \bigr) \\ &\quad\leq e^{-C_{6}t}E_{2} \bigl(\tilde{u}_{0}( \theta_{-t}\omega),v_{0}(\theta_{-t}\omega ), \tilde{\psi}_{0}(\theta_{-t}\omega) \bigr)+\epsilon\int _{0}^{t}e^{C_{6}(s-t)}P_{2}( \theta_{s-t}\omega)\,ds+C_{7} \\ &\quad\leq e^{-C_{6}t}E_{2} \bigl(\tilde{u}_{0}( \theta_{-t}\omega),v_{0}(\theta_{-t}\omega ), \tilde{\psi}_{0}(\theta_{-t}\omega) \bigr)+\epsilon\int _{-t}^{0}e^{C_{6}s}P_{2}( \theta_{s}\omega)\,ds+C_{7} \\ &\quad\leq e^{-C_{6}t}E_{2} \bigl(\tilde{u}_{0}( \theta_{-t}\omega),v_{0}(\theta_{-t}\omega ), \tilde{\psi}_{0}(\theta_{-t}\omega) \bigr)+\epsilon C_{6}\int_{-t}^{0}e^{\frac {C_{6}}{2}s}r( \omega)\,ds+C_{7} \\ &\quad\leq e^{-C_{6}t}E_{2} \bigl(\tilde{u}_{0}( \theta_{-t}\omega),v_{0}(\theta_{-t}\omega ), \tilde{\psi}_{0}(\theta_{-t}\omega) \bigr)+\epsilon C_{6}r(\omega)+C_{7}, \end{aligned}$$
which implies that
$$\begin{aligned} &\bigl\| \tilde{u} \bigl(t,\theta_{-t}\omega, \tilde{u}_{0}(\theta_{-t}\omega) \bigr)\bigr\| ^{2}_{\mathbb{H}^{1}}+ \bigl\| v \bigl(t,\theta_{-t} \omega,v_{0}(\theta_{-t} \omega) \bigr)\bigr\| ^{2}_{\mathbb{H}^{2}} +\bigl\| \tilde{\psi} \bigl(t, \theta_{-t}\omega,\tilde{\psi}_{0}( \theta_{-t} \omega) \bigr)\bigr\| ^{2}_{\mathbb{H}^{1}} \\ &\quad\leq e^{-C_{6}t}E_{2} \bigl(\tilde{u}_{0}( \theta_{-t}\omega),v_{0}(\theta_{-t}\omega ), \tilde{\psi}_{0}(\theta_{-t}\omega) \bigr) +\epsilon C_{6}r(\omega)+C_{7}, \end{aligned}$$
(3.25)
where \(r(\omega)\) is a tempered function.
Note that \(\tilde{u}=uz\), \(\tilde{\psi}=\psi-\epsilon\delta z_{2}(\theta_{t}\omega)\), by (3.25) we obtain
$$\begin{aligned} &\bigl\| u \bigl(t,\theta_{-t}\omega,u_{0}(\theta_{-t} \omega) \bigr)\bigr\| ^{2}_{\mathbb{H}^{1}}+\bigl\| v \bigl(t,\theta_{-t} \omega,v_{0}(\theta_{-t}\omega) \bigr)\bigr\| ^{2}_{\mathbb{H}^{2}} +\bigl\| \psi \bigl(t,\theta_{-t}\omega,\psi_{0}( \theta_{-t}\omega) \bigr)\bigr\| ^{2}_{\mathbb {H}^{1}} \\ &\quad=\bigl\| \tilde{u} \bigl(t,\theta_{-t}\omega,\tilde{u}_{0}( \theta_{-t}\omega )e^{-i\epsilon z_{1}(\theta_{-t}\omega)} \bigr)e^{-i\epsilon z_{1}(\omega)} \bigr\| ^{2}_{\mathbb{H}^{1}} +\bigl\| v \bigl(t,\theta_{-t} \omega,v_{0}(\theta_{-t}\omega) \bigr)\bigr\| ^{2}_{\mathbb{H}^{2}} \\ &\qquad{}+\bigl\| \tilde{\psi} \bigl(t,\theta_{-t}\omega,\tilde{ \psi}_{0}(\theta_{-t}\omega )+\epsilon\delta z_{2}( \theta_{-t}\omega) \bigr)+\epsilon\delta z_{2}(\omega)\bigr\| ^{2}_{\mathbb{H}^{1}} \\ &\quad\leq\bigl\| \tilde{u} \bigl(t,\theta_{-t}\omega,\tilde{u}_{0}( \theta_{-t}\omega )e^{-i\epsilon z_{1}(\theta_{-t}\omega)} \bigr)\bigr\| ^{2}_{\mathbb{H}^{1}}+ \bigl\| \nabla v \bigl(t,\theta_{-t}\omega,v_{0}( \theta_{-t}\omega) \bigr)\bigr\| ^{2}_{\mathbb{H}^{2}} \\ &\qquad{}+\bigl\| \tilde{\psi} \bigl(t,\theta_{-t}\omega,\tilde{ \psi}_{0}(\theta_{-t}\omega )+\epsilon\delta z_{2}( \theta_{-t}\omega) \bigr)\bigr\| ^{2}_{\mathbb{H}^{1}}+2\epsilon ^{2}\|\delta\|^{2}_{\mathbb{H}^{1}}\bigl\| z_{2}(\omega) \bigr\| ^{2}_{\mathbb{H}^{1}} \\ &\quad\leq e^{-C_{6}t}E_{2} \bigl(\tilde{u}_{0}( \theta_{-t}\omega),v_{0}(\theta_{-t}\omega ), \tilde{\psi}_{0}(\theta_{-t}\omega) \bigr)+2 \epsilon^{2}\|\delta\|^{2}_{\mathbb {H}^{1}}\bigl\| z_{2}( \theta_{-t}\omega)\bigr\| ^{2}_{\mathbb{H}^{1}} \\ &\qquad{}+\epsilon C_{6}r(\omega)+C_{7}+2 \epsilon^{2}\|\delta\|^{2}_{\mathbb{H}^{1}}\bigl\| z_{2}( \omega)\bigr\| ^{2}_{\mathbb{H}^{1}}. \end{aligned}$$
Since \(\|\nabla z_{2}(\omega)\|^{2}\) is tempered, there exists \(T_{\mathbb{B}}'(\omega)>T_{1}>0\), for all \(t\geq T_{\mathbb{B}}'\),
$$\begin{aligned} &\bigl\| u \bigl(t,\theta_{-t}\omega,u_{0}(\theta_{-t} \omega) \bigr)\bigr\| ^{2}_{\mathbb{H}^{1}}+\bigl\| v \bigl(t,\theta_{-t} \omega,v_{0}(\theta_{-t}\omega) \bigr)\bigr\| ^{2}_{\mathbb{H}^{2}} +\bigl\| \psi \bigl(t,\theta_{-t}\omega,\psi_{0}( \theta_{-t}\omega) \bigr)\bigr\| ^{2}_{\mathbb {H}^{1}}\leq R(\omega), \end{aligned}$$
where \(R(\omega)=\epsilon C_{6}r(\omega)+C_{7}+2\epsilon^{2}\|\delta\|^{2}_{\mathbb{H}^{1}}\|z_{2}(\omega)\| ^{2}_{\mathbb{H}^{1}}\). This completes the proof. □
In what follows, we will use the method of [11, 16] to derive uniform estimates on the tails of solution when x exists on unbounded domains.
Lemma 3.6
Let
\(f\in\mathbb{L}^{4}\), \(g\in\mathbb{L}^{2}\), \(\delta\in\mathbb{H}^{1}\cap\mathbb{W}^{1,4}\), \(\mathbb{B}=\{B(\omega)\}_{\omega\in\Omega}\in\mathbb{D}\)
and
\((u_{0},v_{0},\psi_{0})\in\mathbb{B}(\omega)\). Assume that
\(\mu-\rho(\nu-\rho)>0\). Then, for every
\(\varepsilon>0\)
and
P-a.e. \(\omega\in\Omega\), there exist
\(T'=T'(\omega,\varepsilon)<0\)
and
\(m=m(\omega,\varepsilon)>0\)
such that the solution of system (3.1) satisfies, for all
\(t\geq T'(\omega,\varepsilon)\),
$$ \int_{|x|\geq m}\bigl|\phi \bigl(t,\theta_{-t} \omega,(u_{0},v_{0},\phi_{0}) ( \theta_{-t}\omega ) \bigr)\bigr|^{2}\,dx\leq\varepsilon. $$
Proof
Let \(\eta(x)\in\mathbb{C}(\mathbb{R}^{+},[0,1])\) be a cut-off function satisfying
$$ \eta(x)=0, \quad \mbox{for all } x\in[0,1]; \qquad \eta(x)=1, \quad\mbox{for all } x\in[2,+\infty), $$
and \(|\eta'(x)|\leq\eta_{0}\) (a positive constant).
Taking the imaginary part of the inner product of the first equation of (3.6) with \(\eta(\frac{|x|^{2}}{m^{2}})\tilde{u}\) in \(\mathbb{L}^{2}(\mathbb{R}^{n})\), we have
$$\begin{aligned} &\frac{d}{dt}\int_{\mathbb{R}^{n}}\eta \biggl( \frac{|x|^{2}}{m^{2}} \biggr)|\tilde {u}|^{2}\,dx+2\alpha\int _{\mathbb{R}^{n}}\eta \biggl(\frac{|x|^{2}}{m^{2}} \biggr)| \tilde{u}|^{2}\,dx =2\operatorname{Im}\int_{\mathbb{R}^{n}}\eta \biggl( \frac{|x|^{2}}{m^{2}} \biggr)fz\tilde{u}\,dx. \end{aligned}$$
(3.26)
Taking the inner product of the third equation of (3.6) with \(\eta(\frac{|x|^{2}}{m^{2}})\tilde{\psi}\) in \(\mathbb{L}^{2}(\mathbb{R}^{n})\), we get
$$\begin{aligned} &\frac{d}{dt}\int_{\mathbb{R}^{n}}\eta \biggl( \frac{|x|^{2}}{m^{2}} \biggr)|\tilde{\psi }|^{2}\,dx+2(\nu-\rho) \int _{\mathbb{R}^{n}}\eta \biggl(\frac{|x|^{2}}{m^{2}} \biggr)|\tilde{ \psi}|^{2}\,dx \\ &\quad=-2 \bigl[\mu-\rho(\nu-\rho) \bigr]\int_{\mathbb{R}^{n}}\eta \biggl( \frac {|x|^{2}}{m^{2}} \biggr)v\tilde{\psi}\,dx +2\int_{\mathbb{R}^{n}} \eta \biggl(\frac{|x|^{2}}{m^{2}} \biggr) (\triangle v)\tilde{\psi}\,dx \\ &\qquad{}+2\beta\int_{\mathbb{R}^{n}}\eta \biggl(\frac{|x|^{2}}{m^{2}} \biggr)\bigl|\tilde {u}z^{-1}\bigr|^{2}\tilde{\psi}\,dx-2\epsilon(\nu- \rho)\int_{\mathbb{R}^{n}}\eta \biggl(\frac{|x|^{2}}{m^{2}} \biggr)\delta z_{2}(\theta_{t}\omega)\tilde{\psi}\,dx \\ &\qquad{}+2\int_{\mathbb{R}^{n}}\eta \biggl(\frac{|x|^{2}}{m^{2}} \biggr)g \tilde{\psi}\,dx. \end{aligned}$$
(3.27)
Note that
$$\begin{aligned} &{-}2 \bigl[\mu-\rho(\nu-\rho) \bigr]\int_{\mathbb{R}^{n}} \eta \biggl(\frac {|x|^{2}}{m^{2}} \biggr)v\tilde{\psi}\,dx \\ &\quad=- \bigl[\mu-\rho(\nu-\rho) \bigr]\frac{d}{dt}\int _{\mathbb{R}^{n}}\eta \biggl(\frac {|x|^{2}}{m^{2}} \biggr)|v|^{2}\,dx- 2\rho \bigl[\mu-\rho(\nu-\rho) \bigr]\int_{\mathbb{R}^{n}}\eta \biggl( \frac {|x|^{2}}{m^{2}} \biggr)|v|^{2}\,dx \\ &\qquad{}+2\epsilon \bigl[\mu-\rho(\nu-\rho) \bigr]\int_{\mathbb{R}^{n}} \eta \biggl(\frac {|x|^{2}}{m^{2}} \biggr)\delta z_{2}( \theta_{t} \omega)v \,dx, \end{aligned}$$
(3.28)
and
$$\begin{aligned} &2\int_{\mathbb{R}^{n}}\eta \biggl(\frac{|x|^{2}}{m^{2}} \biggr) (\triangle v)\tilde{\psi}\,dx \\ &\quad=-2\int_{\mathbb{R}^{n}}\eta' \biggl( \frac{|x|^{2}}{m^{2}} \biggr) \biggl(\frac{2x}{m^{2}} \biggr) (\nabla v) \tilde{ \psi}\,dx-\frac{d}{dt}\int_{\mathbb{R}^{n}}\eta \biggl( \frac {|x|^{2}}{m^{2}} \biggr)|\nabla v|^{2}\,dx \\ &\qquad{}-2\rho\int_{\mathbb{R}^{n}}\eta \biggl(\frac{|x|^{2}}{m^{2}} \biggr)|\nabla v|^{2}\,dx-2\rho\int_{\mathbb{R}^{n}} \eta' \biggl(\frac{|x|^{2}}{m^{2}} \biggr) \biggl(\frac {2x}{m^{2}} \biggr)v\nabla v\,dx \\ &\qquad{}+2\epsilon\int_{\mathbb{R}^{n}}\eta \biggl(\frac{|x|^{2}}{m^{2}} \biggr)z_{2}(\theta_{t}\omega )\nabla v\nabla\delta \,dx+2 \epsilon\int_{\mathbb{R}^{n}}\eta \biggl(\frac{|x|^{2}}{m^{2}} \biggr)\delta \nabla v\nabla z_{2}(\theta_{t}\omega)\,dx \\ &\qquad{}+2\epsilon\int_{\mathbb{R}^{n}}\eta' \biggl( \frac{|x|^{2}}{m^{2}} \biggr) \biggl(\frac {2x}{m^{2}} \biggr)\delta z_{2}( \theta_{-t}\omega)\nabla v. \end{aligned}$$
(3.29)
Summing up (3.26)-(3.29), we have
$$\begin{aligned} &\frac{d}{dt}\int_{\mathbb{R}^{n}}\eta \biggl( \frac{|x|^{2}}{m^{2}} \biggr) \bigl(|\tilde {u}|^{2}+ \bigl(\mu-\rho(\nu- \rho) \bigr)|v|^{2}+|\nabla v|^{2}+|\tilde{ \psi}|^{2} \bigr)\,dx \\ &\qquad{}+2\alpha\int_{\mathbb{R}^{n}}\eta \biggl(\frac{|x|^{2}}{m^{2}} \biggr)|\tilde{u}|^{2}\,dx+ 2\rho \bigl(\mu-\rho(\nu-\rho) \bigr)\int _{\mathbb{R}^{n}}\eta \biggl(\frac{|x|^{2}}{m^{2}} \biggr)|v|^{2}\,dx \\ &\qquad{}+2\rho\int_{\mathbb{R}^{n}}\eta \biggl(\frac{|x|^{2}}{m^{2}} \biggr)|\nabla v|^{2}\,dx+2(\nu -\rho)\int_{\mathbb{R}^{n}}\eta \biggl(\frac{|x|^{2}}{m^{2}} \biggr)|\tilde{\psi }|^{2}\,dx \\ &\quad=2\operatorname{Im}\int_{\mathbb{R}^{n}}\eta \biggl(\frac{|x|^{2}}{m^{2}} \biggr)fz \tilde{u}\,dx+2\epsilon \bigl(\mu-\rho(\nu-\rho) \bigr)\int_{\mathbb{R}^{n}} \eta \biggl(\frac{|x|^{2}}{m^{2}} \biggr)\delta vz_{2}( \theta_{t} \omega)\,dx \\ &\qquad{}-2\int_{\mathbb{R}^{n}}\eta' \biggl( \frac{|x|^{2}}{m^{2}} \biggr) \biggl(\frac{2x}{m^{2}} \biggr) (\nabla v)\tilde{ \psi}\,dx+2\epsilon\int_{\mathbb{R}^{n}}\eta \biggl(\frac {|x|^{2}}{m^{2}} \biggr)z_{2}(\theta_{t}\omega)\nabla v\nabla\delta \,dx \\ &\qquad{}+2\epsilon\int_{\mathbb{R}^{n}}\eta \biggl(\frac{|x|^{2}}{m^{2}} \biggr)\delta\nabla v\nabla z_{2}(\theta_{t}\omega)\,dx-2\rho \int_{\mathbb{R}^{n}}\eta' \biggl(\frac {|x|^{2}}{m^{2}} \biggr) \biggl(\frac{2x}{m^{2}} \biggr)v\nabla v\,dx \\ &\qquad{}+2\epsilon\int_{\mathbb{R}^{n}}\eta' \biggl( \frac{|x|^{2}}{m^{2}} \biggr) \biggl(\frac {2x}{m^{2}} \biggr)\delta z_{2}( \theta_{-t}\omega)\nabla v\,dx +2\beta\int _{\mathbb{R}^{n}}\eta \biggl(\frac{|x|^{2}}{m^{2}} \biggr)\bigl|\tilde {u}z^{-1}\bigr|^{2}\tilde{ \psi}\,dx \\ &\qquad{}+2\int_{\mathbb{R}^{n}}\eta \biggl(\frac{|x|^{2}}{m^{2}} \biggr)g \tilde{\psi}\,dx-2\epsilon (\nu-\rho)\int_{\mathbb{R}^{n}}\eta \biggl( \frac{|x|^{2}}{m^{2}} \biggr)\delta z_{2}(\theta_{t}\omega) \tilde{\psi}\,dx. \end{aligned}$$
(3.30)
We now estimate the right-hand side term in (3.30) as follows. Firstly, by the definition of η, we have
$$\begin{aligned} &{-}2\int_{\mathbb{R}^{n}}\eta' \biggl( \frac{|x|^{2}}{m^{2}} \biggr) \biggl(\frac{2x}{m^{2}} \biggr) (\nabla v)\tilde{ \psi}\,dx \leq4\eta_{0}\int_{m\leq x\leq\sqrt{2}m} \biggl( \frac{|x|}{m^{2}} \biggr)|\nabla v||\tilde{\psi}|\,dx \\ &\hphantom{{-}2\int_{\mathbb{R}^{n}}\eta' \biggl( \frac{|x|^{2}}{m^{2}} \biggr) \biggl(\frac{2x}{m^{2}} \biggr) (\nabla v)\tilde{ \psi}\,dx}\leq\frac{4\sqrt{2}\eta_{0}}{m} \bigl(\|\nabla v\|^{2}+\|\tilde{\psi} \|^{2} \bigr), \end{aligned}$$
(3.31)
$$\begin{aligned} &{-}2\rho\int_{\mathbb{R}^{n}}\eta' \biggl(\frac{|x|^{2}}{m^{2}} \biggr) \biggl(\frac {2x}{m^{2}} \biggr)v\nabla v\,dx \leq4 \rho\eta_{0}\int_{m\leq x \leq\sqrt{2}m} \biggl( \frac{|x|}{m^{2}} \biggr)|v||\nabla v|\,dx \\ &\hphantom{{-}2\rho\int_{\mathbb{R}^{n}}\eta' \biggl(\frac{|x|^{2}}{m^{2}} \biggr) \biggl(\frac {2x}{m^{2}} \biggr)v\nabla v\,dx}\leq\frac{4\rho\sqrt{2}\eta_{0}}{m} \bigl(\|\nabla v\|^{2}+\|v \|^{2} \bigr), \end{aligned}$$
(3.32)
$$\begin{aligned} &2\epsilon\int_{\mathbb{R}^{n}}\eta' \biggl(\frac{|x|^{2}}{m^{2}} \biggr) \biggl(\frac {2x}{m^{2}} \biggr)\delta z_{2}(\theta_{t}\omega)\nabla v\,dx \\ &\quad\leq4\epsilon \eta_{0}\int_{m\leq x\leq\sqrt{2}m} \biggl(\frac{|x|}{m^{2}} \biggr)| \delta|\bigl|z_{2}(\theta_{t}\omega)\bigr||\nabla v|\,dx \\ &\quad\leq\frac{4\epsilon\sqrt{2}\eta_{0}}{m} \bigl(\|\nabla v\|^{2}+\|\delta \|^{2}\bigl\| z_{2}(\theta_{-t}\omega)\bigr\| ^{2} \bigr). \end{aligned}$$
(3.33)
Then, by Young’s inequality, we get
$$\begin{aligned}& 2\operatorname{Im}\int_{\mathbb{R}^{n}}\eta \biggl( \frac{|x|^{2}}{m^{2}} \biggr)fz\tilde{u}\,dx \leq\alpha\int_{\mathbb{R}^{n}}\eta \biggl(\frac{|x|^{2}}{m^{2}} \biggr)|\tilde {u}|^{2}\,dx+\frac{4}{\alpha}\int_{\mathbb{R}^{n}} \eta \biggl(\frac {|x|^{2}}{m^{2}} \biggr)|f|^{2}\,dx; \end{aligned}$$
(3.34)
$$\begin{aligned}& \begin{aligned}[b] &2\epsilon \bigl(\mu-\rho(\nu-\rho) \bigr)\int _{\mathbb{R}^{n}}\eta \biggl(\frac{|x|^{2}}{m^{2}} \biggr) \delta vz_{2}(\theta_{t}\omega)\,dx \\ &\quad\leq\rho \bigl(\mu-\rho(\nu-\rho) \bigr)\int_{\mathbb{R}^{n}}\eta \biggl(\frac{|x|^{2}}{m^{2}} \biggr)|v|^{2}\,dx \\ &\qquad{}+\frac{4\epsilon^{2}}{\rho(\mu-\rho(\nu-\rho))}\int_{\mathbb{R}^{n}}\eta \biggl( \frac{|x|^{2}}{m^{2}} \biggr)|\delta|^{2}\bigl|z_{2}( \theta_{t}\omega)\bigr|^{2}\,dx; \end{aligned} \end{aligned}$$
(3.35)
$$\begin{aligned}& \begin{aligned}[b] &2\epsilon\int_{\mathbb{R}^{n}}\eta \biggl( \frac{|x|^{2}}{m^{2}} \biggr)z_{2}(\theta_{t}\omega )\nabla v \nabla\delta \,dx \\ &\quad\leq\frac{\rho}{2}\int_{\mathbb{R}^{n}}\eta \biggl( \frac{|x|^{2}}{m^{2}} \biggr)|\nabla v|^{2}\,dx +\frac{8\epsilon^{2}}{\rho}\int _{\mathbb{R}^{n}}\eta \biggl(\frac {|x|^{2}}{m^{2}} \biggr)\bigl|z_{2}( \theta_{t}\omega)\bigr|^{2}|\nabla\delta|^{2}\,dx; \end{aligned} \end{aligned}$$
(3.36)
$$\begin{aligned}& \begin{aligned}[b] &2\epsilon\int_{\mathbb{R}^{n}}\eta \biggl( \frac{|x|^{2}}{m^{2}} \biggr)\delta\nabla z_{2}(\theta_{t} \omega) \nabla v\,dx \\ &\quad\leq\frac{\rho}{2}\int_{\mathbb{R}^{n}}\eta \biggl( \frac{|x|^{2}}{m^{2}} \biggr)|\nabla v|^{2}\,dx +\frac{8\epsilon^{2}}{\rho}\int _{\mathbb{R}^{n}}\eta \biggl(\frac {|x|^{2}}{m^{2}} \biggr)\bigl|\nabla z_{2}(\theta_{t}\omega)\bigr|^{2}|\delta|^{2}\,dx; \end{aligned} \end{aligned}$$
(3.37)
$$\begin{aligned}& \begin{aligned}[b] &2\beta\int_{\mathbb{R}^{n}}\eta \biggl( \frac{|x|^{2}}{m^{2}} \biggr)\bigl|\tilde {u}z^{-1}\bigr|^{2}\tilde{\psi}\,dx \\ &\quad\leq\frac{\nu-\rho}{2}\int_{\mathbb{R}^{n}}\eta \biggl( \frac {|x|^{2}}{m^{2}} \biggr)|\tilde{\psi}|^{2}\,dx +\frac{8\beta^{2}}{\nu-\rho}\int _{\mathbb{R}^{n}}\eta \biggl(\frac {|x|^{2}}{m^{2}} \biggr)| \tilde{u}|^{4}\,dx; \end{aligned} \end{aligned}$$
(3.38)
$$\begin{aligned}& \begin{aligned}[b] &{-}2\epsilon(\nu-\rho)\int_{\mathbb{R}^{n}}\eta \biggl(\frac{|x|^{2}}{m^{2}} \biggr)\delta z_{2}(\theta_{t} \omega)\tilde{\psi}\,dx \\ &\quad\leq\frac{\nu-\rho}{2}\int_{\mathbb{R}^{n}}\eta \biggl( \frac {|x|^{2}}{m^{2}} \biggr)|\tilde{\psi}|^{2}\,dx+8\epsilon^{2}( \nu-\rho)\int_{\mathbb {R}^{n}}\eta \biggl(\frac{|x|^{2}}{m^{2}} \biggr)\bigl|z_{2}(\theta_{t}\omega)\bigr|^{2}| \delta|^{2}\,dx; \end{aligned} \end{aligned}$$
(3.39)
$$\begin{aligned}& \begin{aligned}[b] &2\int_{\mathbb{R}^{n}}\eta \biggl(\frac{|x|^{2}}{m^{2}} \biggr)g\tilde{\psi}\,dx \\ &\quad\leq\frac{\nu-\rho}{2}\int_{\mathbb{R}^{n}}\eta \biggl( \frac {|x|^{2}}{m^{2}} \biggr)|\tilde{\psi}|^{2}\,dx +\frac{8}{\nu-\rho}\int _{\mathbb{R}^{n}}\eta \biggl(\frac{|x|^{2}}{m^{2}} \biggr)|g|^{2}\,dx. \end{aligned} \end{aligned}$$
(3.40)
Finally, by (3.30)-(3.40), we obtain
$$\begin{aligned} &\frac{d}{dt}\int_{\mathbb{R}^{n}}\eta \biggl( \frac{|x|^{2}}{m^{2}} \biggr) \bigl(|\tilde {u}|^{2}+ \bigl(\mu-\rho(\nu- \rho) \bigr)|v|^{2}+|\nabla v|^{2}+|\tilde{ \psi}|^{2} \bigr)\,dx \\ &\qquad{}+\alpha\int_{\mathbb{R}^{n}}\eta \biggl(\frac{|x|^{2}}{m^{2}} \biggr)|\tilde{u}|^{2}\,dx +\rho \bigl(\mu-\rho(\nu-\rho) \bigr)\int _{\mathbb{R}^{n}}\eta \biggl(\frac {|x|^{2}}{m^{2}} \biggr)|v|^{2}\,dx \\ &\qquad{}+\rho\int_{\mathbb{R}^{n}}\eta \biggl(\frac{|x|^{2}}{m^{2}} \biggr)| \nabla v|^{2}\,dx+\frac {\nu-\rho}{2} \int_{\mathbb{R}^{n}} \eta \biggl(\frac{|x|^{2}}{m^{2}} \biggr)|\tilde{\psi}|^{2}\,dx \\ &\quad\leq\frac{4}{\alpha}\int_{\mathbb{R}^{n}}\eta \biggl( \frac{|x|^{2}}{m^{2}} \biggr)|f|^{2}\,dx +\frac{4\epsilon^{2}}{\rho(\mu-\rho(\nu-\rho))}\int _{\mathbb{R}^{n}} \eta \biggl(\frac{|x|^{2}}{m^{2}} \biggr)| \delta|^{2}\bigl|z_{2}(\theta_{t}\omega)\bigr|^{2}\,dx \\ &\qquad{}+\frac{8\epsilon^{2}}{\rho}\int_{\mathbb{R}^{n}}\eta \biggl( \frac{|x|^{2}}{m^{2}} \biggr) \bigl(\bigl|z_{2}(\theta_{t} \omega)\bigr|^{2}|\nabla\delta|^{2}+\bigl|\nabla z_{2}( \theta_{t}\omega)\bigr|^{2}|\delta|^{2} \bigr)\,dx \\ &\qquad{}+\frac{8\beta^{2}}{\nu-\rho}\int_{\mathbb{R}^{n}}\eta \biggl( \frac {|x|^{2}}{m^{2}} \biggr)|\tilde{u}|^{4}\,dx +8\epsilon^{2}( \nu- \rho)\int_{\mathbb{R}^{n}}\eta \biggl(\frac {|x|^{2}}{m^{2}} \biggr)\bigl|z_{2}(\theta_{t}\omega)\bigr|^{2}| \delta|^{2}\,dx \\ &\qquad{}+\frac{8}{\nu-\rho}\int_{\mathbb{R}^{n}}\eta \biggl( \frac {|x|^{2}}{m^{2}} \biggr)|g|^{2}\,dx+\frac{4\sqrt{2}\eta_{0}}{m} \bigl(\|\nabla v \|^{2}+\|\tilde{\psi}\|^{2} \bigr) \\ &\qquad{}+\frac{4\rho\sqrt{2}\eta_{0}}{m} \bigl(\|\nabla v\|^{2}+\|v \|^{2} \bigr)+\frac{4\epsilon\sqrt{2}\eta_{0}}{m} \bigl(\|\nabla v\|^{2}+\| \delta\|^{2} \bigl\| z_{2}(\theta_{-t}\omega) \bigr\| ^{2} \bigr). \end{aligned}$$
(3.41)
By (3.41) and assumption, there exist positive constants \(C_{8}:=\min\{\alpha,\rho,\frac{\nu-\rho}{2}\}\) and \(C_{9}:=C_{9}(\alpha,\beta,\mu,\nu,\rho)\) such that
$$\begin{aligned} &\frac{d}{dt}\int_{\mathbb{R}^{n}}\eta \biggl( \frac{|x|^{2}}{m^{2}} \biggr) \bigl(|\tilde {u}|^{2}+ \bigl(\mu-\rho(\nu- \rho) \bigr)|v|^{2}+|\nabla v|^{2}+|\tilde{ \psi}|^{2} \bigr)\,dx \\ &\qquad{}+C_{8}\int_{\mathbb{R}^{n}}\eta \biggl( \frac{|x|^{2}}{m^{2}} \biggr) \bigl(|\tilde{u}|^{2}+\rho \bigl(\mu -\rho( \nu- \rho) \bigr)|v|^{2}+|\nabla v|^{2}+|\tilde{ \psi}|^{2} \bigr)\,dx \\ &\quad\leq\epsilon C_{9}\int_{\mathbb{R}^{n}}\eta \biggl( \frac{|x|^{2}}{m^{2}} \biggr) \bigl(\bigl|z_{2}(\theta_{t}\omega )\bigr|^{4}+\bigl|\nabla z_{2}(\theta_{t} \omega)\bigr|^{4} \bigr)\,dx \\ &\qquad{}+C_{9}\int_{\mathbb{R}^{n}}\eta \biggl( \frac{|x|^{2}}{m^{2}} \biggr) \bigl(|f|^{4}+|g|^{2}+|\delta |^{4}+|\nabla\delta|^{4} \bigr)\,dx +C_{9}\int _{\mathbb{R}^{n}}\eta \biggl(\frac{|x|^{2}}{m^{2}} \biggr)| \tilde{u}|^{4}\,dx \\ &\qquad{}+\frac{4\sqrt{2}\eta_{0}}{m} \bigl(\|\nabla v\|^{2}+\|\tilde{\psi} \|^{2} \bigr)+\frac{4\epsilon\sqrt{2}\eta_{0}}{m} \bigl(\|\nabla v\|^{2}+\| \delta \|^{2}\bigl\| z_{2}(\theta_{-t}\omega) \bigr\| ^{2} \bigr). \end{aligned}$$
(3.42)
Now, replacing ω by \(\theta_{-t}\omega\), and then integrating (3.42) over \((T_{2},t)\) with \(t\geq T_{2}\), we have
$$\begin{aligned} &\int_{\mathbb{R}^{n}}\eta \biggl(\frac{|x|^{2}}{m^{2}} \biggr) \bigl(\bigl|\tilde{u} \bigl(t,\theta_{-t}\omega,\tilde{u}_{0}( \theta_{-t}\omega) \bigr)\bigr|^{2} + \bigl(\mu-\rho(\nu-\rho) \bigr)\bigl|v \bigl(t,\theta_{-t}\omega,v_{0}( \theta_{-t} \omega ) \bigr)\bigr|^{2} \\ &\qquad{}+\bigl|\nabla v \bigl(t,\theta_{-t}\omega,v_{0}( \theta_{-t}\omega) \bigr)\bigr|^{2} +\bigl|\tilde{\psi} \bigl(t, \theta_{-t}\omega,\tilde{\psi}_{0}(\theta_{-t} \omega ) \bigr)\bigr|^{2} \bigr)\,dx \\ &\quad\leq e^{-C_{8}(t-T_{2})}\int_{\mathbb{R}^{n}}\eta \biggl( \frac{|x|^{2}}{m^{2}} \biggr) \bigl(\bigl|\tilde{u} \bigl(T_{2}, \theta_{-t}\omega,\tilde{u}_{0}(\theta_{-t}\omega) \bigr)\bigr|^{2} +\bigl|\nabla v \bigl(T_{2},\theta_{-t} \omega,v_{0}(\theta_{-t}\omega) \bigr)\bigr|^{2} \\ &\qquad{}+\rho \bigl(\mu-\rho(\nu-\rho) \bigr)\bigl|v \bigl(T_{2}, \theta_{-t}\omega,v_{0}(\theta _{-t}\omega) \bigr)\bigr|^{2}+\bigl|\tilde{\psi} \bigl(T_{2},\theta_{-t} \omega,\tilde{\psi }_{0}(\theta_{-t}\omega) \bigr)\bigr|^{2} \bigr)\,dx \\ &\qquad{}+\epsilon C_{9}\int_{T_{2}}^{t}e^{C_{8}(s-t)} \int_{\mathbb{R}^{n}}\eta \biggl(\frac {|x|^{2}}{m^{2}} \biggr) \bigl(\bigl|z_{2}(\theta_{s-t}\omega)\bigr|^{4}+\bigl|\nabla z_{2}(\theta_{s-t}\omega )\bigr|^{4} \bigr)\,dx\,ds \\ &\qquad{}+C_{9}\int_{T_{2}}^{t}e^{C_{8}(s-t)} \int_{\mathbb{R}^{n}}\eta \biggl(\frac {|x|^{2}}{m^{2}} \biggr) \bigl(|f|^{4}+|g|^{2}+|\delta|^{4}+|\nabla \delta|^{4}+|\tilde {u}|^{4} \bigr)\,dx\,ds \\ &\qquad{}+\frac{C^{*}}{m}\int_{T_{2}}^{t}e^{C_{8}(s-t)} \bigl(\bigl\| v \bigl(s,\theta_{-t}\omega ,v_{0}( \theta_{-t}\omega) \bigr)\bigr\| ^{2}+\bigl\| \nabla v \bigl(s, \theta_{-t}\omega,v_{0}(\theta_{-t}\omega) \bigr) \bigr\| ^{2} \\ &\qquad{}+\bigl\| \tilde{\psi} \bigl(s,\theta_{-t}\omega,\tilde{ \psi}_{0}(\theta_{-t}\omega ) \bigr)\bigr\| ^{2} \bigr)\,ds +\frac{4\epsilon\sqrt{2}\eta_{0}\|\delta\|^{2}}{m}\int_{T_{2}}^{t}e^{C_{8}(s-t)} \bigl\| z_{2}(\theta_{-t}\omega)\bigr\| ^{2}\,ds, \end{aligned}$$
(3.43)
where \(C^{*}\) is a fixed constant.
In what follows, we estimate the terms in (3.43). First replacing t by \(T_{2}\) and then replacing ω by \(\theta_{-t}\omega\) in (3.18), we have the following bounds for the first term on the right-hand side of (3.43):
$$\begin{aligned} &e^{-C_{8}(t-T_{2})}\int_{\mathbb{R}^{n}}\eta \biggl( \frac{|x|^{2}}{m^{2}} \biggr) \bigl(\bigl|\tilde{u} \bigl(T_{2}, \theta_{-t}\omega,\tilde{u}_{0}(\theta_{-t}\omega) \bigr)\bigr|^{2} +\bigl|\nabla v \bigl(T_{2},\theta_{-t} \omega,v_{0}(\theta_{-t}\omega) \bigr)\bigr|^{2} \\ &\qquad{}+\rho \bigl(\mu-\rho(\nu-\rho) \bigr)\bigl|v \bigl(T_{2}, \theta_{-t}\omega,v_{0}(\theta _{-t}\omega) \bigr)\bigr|^{2} +\bigl|\tilde{\psi} \bigl(T_{2},\theta_{-t} \omega,\tilde{\psi}_{0}(\theta_{-t}\omega ) \bigr)\bigr|^{2} \bigr)\,dx \\ &\quad\leq e^{-C_{8}(t-T_{2})} \bigl(e^{-C_{8}T_{2}}\bigl\| \tilde{u}_{0}( \theta_{-t}\omega)\bigr\| ^{2} +\rho \bigl(\mu-\rho(\nu-\rho) \bigr) \bigl\| v_{0}(\theta_{-t}\omega)\bigr\| ^{2} \\ &\qquad{}+\bigl\| \tilde{\psi}_{0}(\theta_{-t}\omega) \bigr\| ^{2} \bigr) +e^{-C_{8}(t-T_{2})}\int_{0}^{T_{2}}e^{C_{8}(s-T_{2})} \bigl(P_{1}(\theta_{s-t}\omega )+C_{9} \bigr)\,ds \\ &\quad\leq e^{-C_{8}t} \bigl(\bigl\| \tilde{u}_{0}( \theta_{-t}\omega)\bigr\| ^{2} +\rho \bigl(\mu-\rho(\nu-\rho) \bigr) \bigl\| v_{0}(\theta_{-t}\omega)\bigr\| ^{2}+\bigl\| \tilde{\psi }_{0}(\theta_{-t}\omega)\bigr\| ^{2} \bigr) \\ &\qquad{}+\int_{-t}^{T_{2}-t}e^{C_{8}s}P_{1}( \theta_{s}\omega )\,ds+C_{9}e^{C_{8}(T_{2}-t)} \\ &\quad\leq e^{-C_{8}t} \bigl(\bigl\| \tilde{u}_{0}( \theta_{-t}\omega)\bigr\| ^{2} +\rho \bigl(\mu-\rho(\nu-\rho) \bigr) \bigl\| v_{0}(\theta_{-t}\omega)\bigr\| ^{2}+\bigl\| \tilde{\psi }_{0}(\theta_{-t}\omega)\bigr\| ^{2} \bigr) \\ &\qquad{}+C_{9}\int_{-t}^{T_{2}-t}e^{\frac{C_{8}}{2}s}r( \omega )\,ds+C_{9}e^{C_{8}(T_{2}-t)} \\ &\quad\leq e^{-C_{8}t} \bigl(\bigl\| \tilde{u}_{0}( \theta_{-t}\omega)\bigr\| ^{2} +\rho \bigl(\mu-\rho(\nu-\rho) \bigr) \bigl\| v_{0}(\theta_{-t}\omega)\bigr\| ^{2}+\bigl\| \tilde{\psi }_{0}(\theta_{-t}\omega)\bigr\| ^{2} \bigr) \\ &\qquad{}+\frac{2C_{9}}{C_{8}}e^{\frac{C_{8}}{2}(T_{2}-t)}r(\omega)+C_{9}e^{C_{8}(T_{2}-t)}. \end{aligned}$$
(3.44)
By (3.44) we find that, given \(\varepsilon>0\), there is \(T_{3}=T_{3}(\mathbb{B},\omega,\varepsilon)>T_{2}\) such that for all \(t>T_{3}\),
$$\begin{aligned} &e^{-C(t-T_{2})}\int_{\mathbb{R}^{n}}\eta \biggl( \frac{|x|^{2}}{m^{2}} \biggr) \bigl(\bigl|\tilde {u} \bigl(T_{2}, \theta_{-t}\omega,\tilde{u}_{0}(\theta_{-t}\omega) \bigr)\bigr|^{2} +\bigl|\nabla v \bigl(T_{2},\theta_{-t} \omega,v_{0}(\theta_{-t}\omega) \bigr)\bigr|^{2} \\ &\qquad{}+\rho \bigl(\mu-\rho(\nu-\rho) \bigr)\bigl|v \bigl(T_{2}, \theta_{-t}\omega,v_{0}(\theta _{-t}\omega) \bigr)\bigr|^{2} +\bigl|\tilde{\psi} \bigl(T_{2},\theta_{-t} \omega,\tilde{\psi}_{0}(\theta_{-t}\omega ) \bigr)\bigr|^{2} \bigr)\,dx \\ &\quad\leq\varepsilon. \end{aligned}$$
(3.45)
Note that \(\delta\in\mathbb{H}^{1}\cap\mathbb{W}^{1,4}\), hence there is \(R_{1}=R_{1}(\omega,\varepsilon)\) such that for all \(m\geq R_{1}\),
$$ \int_{|x|\geq m} \bigl(\bigl|\nabla \delta(x)\bigr|^{4}+\bigl| \delta(x)\bigr|^{4} \bigr)\,dx\leq\frac{\varepsilon}{r(\omega)}, $$
(3.46)
where \(r(\omega)\) is a tempered function. By (3.46), we have the following estimate:
$$\begin{aligned} &C_{9}\int_{T_{2}}^{t}e^{C_{8}(s-t)} \int_{\mathbb{R}^{n}}\eta \biggl(\frac {|x|^{2}}{m^{2}} \biggr) \bigl(\bigl|z_{2}(\theta_{s-t}\omega)\bigr|^{4} +\bigl|\nabla z_{2}(\theta_{s-t}\omega)\bigr|^{4} \bigr)\,dx\,ds \\ &\quad\leq C_{9}\int_{T_{2}}^{t}e^{C_{8}(s-t)} \int_{|x|\geq m} \bigl(\bigl|z_{2}(\theta _{s-t} \omega)\bigr|^{4} +\bigl|\nabla z_{2}(\theta_{s-t} \omega)\bigr|^{4} \bigr)\,dx\,ds \\ &\quad\leq C_{9}\int_{T_{2}}^{t}e^{C_{8}(s-t)} \int_{|x|\geq m} \bigl(|\delta |^{4}\bigl|y_{2}( \theta_{s-t}\omega_{2})\bigr|^{4} +|\nabla \delta|^{4}\bigl|\nabla y_{2}(\theta_{s-t} \omega_{2})\bigr|^{4} \bigr)\,dx\,ds \\ &\quad\leq\frac{C_{9}\varepsilon}{r(\omega)}\int_{T_{2}}^{t}e^{C_{8}(s-t)} \sum_{i=1}^{2} \bigl(\bigl|y_{i}( \theta_{s-t}\omega_{i})\bigr|^{2} +\bigl|y_{i}( \theta_{s-t}\omega_{i})\bigr|^{4} \bigr)\,ds \\ &\quad\leq\frac{C_{9}\varepsilon}{r(\omega)} \int_{T_{2}}^{t}e^{C_{8}(s-t)}r( \theta _{s-t}\omega)\,ds \leq\frac{C_{9}\varepsilon}{r(\omega)}\int_{T_{2}-t}^{0}e^{C_{8}s}r( \theta _{s}\omega)\,ds \\ &\quad\leq\frac{C_{9}\varepsilon}{r(\omega)}\int_{0}^{T_{2}-t}e^{\frac {C_{8}}{2}s}r( \omega)\,ds\leq\varepsilon. \end{aligned}$$
(3.47)
By Lemma 3.3, we have
$$\begin{aligned} &C_{9}\int_{T_{2}}^{t}e^{C_{8}(s-t)} \int_{\mathbb{R}^{n}}\eta \biggl(\frac {|x|^{2}}{m^{2}} \biggr)| \tilde{u}|^{4}\,dx\,ds \\ &\quad\leq C_{9}\int_{T_{2}}^{t}e^{C_{8}(s-t)}e^{-\alpha s} \int_{|x|\geq m}|\tilde{u}_{0}|^{4}\,dx\,ds + \frac{64C_{9}}{\alpha^{4}}\int_{T_{2}}^{t}e^{C_{8}(s-t)} \int_{|x|\geq m}|f|^{4}\,dx\,ds \\ &\quad\leq\frac{C_{9}}{C_{9}+\alpha}e^{-\alpha t}\int_{|x|\geq m}| \tilde{u}_{0}|^{4}\,dx\,ds+\frac{64C_{9}}{\alpha^{4}}\int _{T_{2}}^{t}e^{C_{8}(s-t)}\int_{|x|\geq m}|f|^{4}\,dx\,ds. \end{aligned}$$
(3.48)
Note that \(f\in\mathbb{L}^{4}\), \(g\in\mathbb{L}^{2}\) and \(\delta\in\mathbb{H}^{1}\cap\mathbb{W}^{1,4}\), there is \(R_{2}=R_{2}(\varepsilon)\) such that for all \(m\geq R_{2}\),
$$ \int_{|x|\geq m} \bigl(|f|^{4}+|g|^{2}+| \delta|^{4}+|\nabla\delta|^{2} \bigr)\,dx\leq\varepsilon. $$
(3.49)
Then, by (3.48) and (3.49), there is \(T_{4}=T_{4}(\mathbb{B},\omega,\varepsilon)>0\) such that for all \(t>T_{4}\),
$$\begin{aligned} &C_{9}\int_{T_{1}}^{t}e^{C_{8}(s-t)} \int_{\mathbb{R}^{n}}\eta \biggl(\frac {|x|^{2}}{m^{2}} \biggr) \bigl(|f|^{4}+|g|^{2}+|\delta|^{2}+|\nabla \delta|^{4}+|\tilde {u}|^{4} \bigr)\,dx\,ds \\ &\quad\leq C_{9}\int_{T_{1}}^{t}e^{C_{8}(s-t)} \int_{|x|\geq m} \bigl(|f|^{4}+|g|^{2}+| \delta|^{2}+|\nabla\delta|^{4} \bigr)\,dx\,ds \\ &\qquad{}+C_{9}e^{-\alpha t}\int_{|x|\geq m}| \tilde{u}_{0}|^{4}\,dx\,ds \\ &\quad\leq C_{9}\varepsilon\int_{T_{1}}^{t}e^{C_{8}(s-t)}\,ds+C_{9}e^{-\alpha t} \int_{|x|\geq m}|\tilde{u}_{0}|^{4}\,dx\,ds\leq \varepsilon. \end{aligned}$$
(3.50)
Now, we estimate the last term on the right-hand side of (3.43). Denote \(E_{3}(v,\tilde{\psi})=\|v\|^{2}+\|\nabla v\|^{2}+\|\tilde{\psi}\|^{2}\). Replacing t by s and then replacing ω by \(\theta_{-t}\omega\) in (3.18), we have the following estimate:
$$\begin{aligned} &\frac{C^{*}}{m}\int_{T_{2}}^{t}e^{C_{8}(s-t)} E_{3}\bigl(v \bigl(s,\theta_{-t}\omega,v_{0}( \theta_{-t}\omega) \bigr),\tilde{\psi} \bigl(s,\theta _{-t} \omega,\tilde{\psi}_{0}(\theta_{-t}\omega) \bigr)\bigr)\,ds \\ &\quad\leq\frac{C^{*}}{m}\int_{T_{2}}^{t}e^{-C_{8}t}E_{1} \bigl(\tilde{u}_{0}(\theta _{-t}\omega),v_{0}( \theta_{-t}\omega),\tilde{\psi}_{0}(\theta_{-t} \omega) \bigr)\,ds \\ &\qquad{}+\frac{C^{*}}{m}\int_{T_{2}}^{t}e^{C_{8}(s-t)} \int_{0}^{s}e^{C_{8}(\tau -s)}P_{1}( \theta_{\tau-t}\omega)\,d\tau \,ds +\frac{C^{*}}{m}\int_{T_{2}}^{t}e^{C_{8}(s-t)}\,ds \\ &\quad\leq\frac{C^{*}}{m}e^{-C_{8}t}(t-T_{2}) E_{1} \bigl(\tilde{u}_{0}(\theta_{-t}\omega),v_{0}( \theta_{-t}\omega),\tilde{\psi }_{0}(\theta_{-t} \omega) \bigr) \\ &\qquad{}+\frac{C^{*}}{m}\int_{T_{2}}^{t}\int _{0}^{s}e^{C_{8}(\tau-t)}P_{1}(\theta _{\tau-t}\omega)\,d\tau \,ds+\frac{C^{*}}{m} \\ &\quad\leq\frac{C^{*}}{m}e^{-C_{8}t}(t-T_{2}) E_{1} \bigl(\tilde{u}_{0}(\theta_{-t}\omega),v_{0}( \theta_{-t}\omega),\tilde{\psi }_{0}(\theta_{-t} \omega) \bigr) \\ &\qquad{}+\frac{C^{*}}{m}\int_{T_{2}}^{t}\int ^{s-t}_{-t}e^{C_{8}\tau}P_{1}( \theta_{\tau }\omega)\,d\tau \,ds+\frac{C^{*}}{m} \\ &\quad\leq\frac{C^{*}}{m}e^{-C_{8}t}(t-T_{2}) E_{1} \bigl(\tilde{u}_{0}(\theta_{-t}\omega),v_{0}( \theta_{-t}\omega),\tilde{\psi }_{0}(\theta_{-t} \omega) \bigr) \\ &\qquad{}+\frac{C^{*}}{m}r(\omega)\int_{T_{2}}^{t} \int^{s-t}_{-t}e^{\frac {C_{8}}{2}\tau}\,d\tau \,ds+ \frac{C^{*}}{m} \\ &\quad\leq\frac{C^{*}}{m}e^{-C_{8}t}(t-T_{2}) E_{1} \bigl(\tilde{u}_{0}(\theta_{-t}\omega),v_{0}( \theta_{-t}\omega),\tilde{\psi }_{0}(\theta_{-t} \omega) \bigr) +\frac{C^{*}}{m}r(\omega)+\frac{C^{*}}{m}, \end{aligned}$$
(3.51)
which implies that there exist \(T_{5}=T_{5}(\mathbb{B},\omega,\varepsilon)>T_{2}\) and \(R_{3}=R_{3}(\omega,\varepsilon)\) such that for all \(t\geq T_{5}\) and \(m\geq R_{3}\),
$$ \frac{C^{*}}{m}\int_{T_{2}}^{t}e^{C_{8}(s-t)}E_{3}\bigl(v \bigl(s,\theta_{-t}\omega ,v_{0}(\theta_{-t} \omega) \bigr),\tilde{\psi} \bigl(s,\theta_{-t}\omega,\tilde{\psi }_{0}(\theta_{-t}\omega) \bigr)\bigr)\,ds \leq\varepsilon. $$
(3.52)
Obviously, there exists \(R_{4}=R_{4}(\omega,\epsilon)\) such that for all \(m>R_{4}\), we have
$$ \frac{4\epsilon\sqrt{2}\eta_{0}\|\delta\|^{2}}{m}\int_{T_{2}}^{t}e^{C_{8}(s-t)} \bigl\| z_{2}(\theta_{-t}\omega)\bigr\| ^{2}\,ds\leq\varepsilon. $$
(3.53)
By (3.43), (3.45), (3.47), (3.50), (3.52) and (3.53), there exist \(R=\max\{R_{1},R_{2},R_{3},R_{4}\}\) and \(T=\min\{T_{2},T_{3},T_{4},T_{5}\}\) such that for all \(m\geq R\) and \(t\geq T\),
$$\begin{aligned} &\int_{\mathbb{R}^{n}}\eta \biggl(\frac{|x|^{2}}{m^{2}} \biggr) \bigl(\bigl|\tilde{u} \bigl(t,\theta _{-t}\omega,\tilde{u}_{0}( \theta_{-t}\omega) \bigr)\bigr|^{2}+ \bigl(\mu-\rho(\nu-\rho ) \bigr)\bigl|v \bigl(t,\theta_{-t}\omega,v_{0}( \theta_{-t} \omega) \bigr)\bigr|^{2} \\ &\quad{}+\bigl|\tilde{\psi} \bigl(t,\theta_{-t}\omega,\tilde{ \psi}_{0}(\theta_{-t}\omega ) \bigr)\bigr|^{2}\bigr)\leq5 \varepsilon, \end{aligned}$$
which shows that for all \(m\geq R\) and \(t\geq T\),
$$\begin{aligned} &\int_{|x|\geq\sqrt{2}m} \bigl(\bigl|\tilde{u} \bigl(t,\theta_{-t} \omega,\tilde {u}_{0}(\theta_{-t}\omega) \bigr)\bigr|^{2}+\bigl|v \bigl(t,\theta_{-t}\omega,v_{0}( \theta _{-t}\omega) \bigr)\bigr|^{2} \\ &\quad{}+\bigl|\tilde{\psi} \bigl(t,\theta_{-t}\omega,\tilde{ \psi}_{0}(\theta_{-t}\omega) \bigr)\bigr|^{2} \bigr) \leq5 \varepsilon. \end{aligned}$$
Note that
$$ \int_{|x|\geq\sqrt{2}m}\bigl|z_{2}(\omega)\bigr|^{2}\,dx=\int _{|x|\geq\sqrt{2}m}|\delta |^{2}\bigl|y_{2}( \omega_{2})\bigr|^{2}\,dx \leq\frac{\varepsilon}{r(\omega)}\bigl|y_{2}( \omega_{2})\bigr|^{2}\leq\varepsilon. $$
Therefore
$$\begin{aligned} &\int_{|x|\geq\sqrt{2}m}\bigl|\phi \bigl(t,\theta_{-t} \omega,(u_{0},v_{0},\phi _{0}) ( \theta_{-t}\omega) \bigr)\bigr|^{2}\,dx \\ &\quad=\int_{|x|\geq\sqrt{2}m} \bigl(\bigl|u \bigl(t,\theta_{-t} \omega,u_{0}(\theta_{-t}\omega ) \bigr)\bigr|^{2}+\bigl|v \bigl(t,\theta_{-t}\omega,v_{0}(\theta_{-t}\omega) \bigr)\bigr|^{2} \\ &\qquad{}+\bigl|\psi \bigl(t,\theta_{-t}\omega,\psi_{0}( \theta_{-t}\omega) \bigr)\bigr|^{2} \bigr)\,dx \\ &\quad=\int_{|x|\geq\sqrt{2}m} \bigl(\bigl|\tilde{u} \bigl(t, \theta_{-t}\omega,\tilde {u}_{0}(\theta_{-t}\omega) \bigr)e^{i\epsilon z_{1}(\omega)}\bigr|^{2}+\bigl|v \bigl(t,\theta_{-t} \omega,v_{0}(\theta_{-t}\omega) \bigr)\bigr|^{2} \\ &\qquad{}+\bigl|\tilde{\psi} \bigl(t,\theta_{-t}\omega,\tilde{ \psi}_{0}(\theta_{-t}\omega ) \bigr)+\epsilon\delta z_{2}(\omega)\bigr|^{2} \bigr) \\ &\quad\leq2\int_{|x|\geq\sqrt{2}m} \bigl(\bigl|\tilde{u} \bigl(t, \theta_{-t}\omega,\tilde {u}_{0}(\theta_{-t}\omega) \bigr)\bigr|^{2}+\bigl|v \bigl(t,\theta_{-t}\omega,v_{0}( \theta _{-t}\omega) \bigr)\bigr|^{2} \\ &\qquad{}+\bigl|\tilde{\psi} \bigl(t,\theta_{-t}\omega,\tilde{ \psi}_{0}(\theta_{-t}\omega ) \bigr)\bigr|^{2}+2 \epsilon^{2}|\delta|^{2}\bigl|z_{2}(\omega)\bigr|^{2} \bigr) \\ &\quad\leq C'\varepsilon, \end{aligned}$$
where \(C'\) is a fixed positive constant. This completes the proof. □
Theorem 3.7
Let
\(f\in\mathbb{H}^{2}\), \(g\in\mathbb{L}^{2}\)
and
\(\delta\in\mathbb{H}\cap\mathbb{W}^{2,4}\). Assume that
\(\alpha(\nu-\rho)>12M_{1}\beta^{2}\), \(\alpha(\mu-\rho(\nu-\rho))>8M_{1}\)
and
\(\mu-\rho(\nu-\rho)>0\). Then the random dynamical system
\(\phi_{\epsilon}\)
possesses a unique
\(\mathbb{D}\)-random attractor in
\(\mathbb{I}\).
Proof
From Proposition 2.8, we only need to prove the asymptotic compactness of RDS ϕ. By Lemma 3.4, for any sequence \(t_{n}\rightarrow\infty\) and \(x_{n}(\theta_{-t_{n}}\omega)\in\mathbb{K}(\omega)\), we have
$$ \bigl\{ \phi(t_{n},\theta_{-t_{n}} \omega,x_{n}) \bigr\} _{n=1}^{\infty} \mbox{ is bounded in } \mathbb{I}. $$
(3.54)
Hence, by (3.54), there exists \(\phi_{0}\in L^{2}(\mathbb{R}^{n})\) such that
$$ \bigl\{ \phi \bigl(t_{n},\theta_{-t_{n}} \omega,x_{n}(\theta_{-t_{n}}\omega) \bigr) \bigr\} _{n=1}^{\infty }\rightarrow\phi_{0} \mbox{ weakly in } \mathbb{I}. $$
(3.55)
Next, we prove that (3.55) is actually strong convergence. Define the set S by \(S=\{x\in\mathbb{R}^{n}: |x|\leq m\}\), where m will be specified latter. Notice the compactness of embedding \(H^{1}(S), H^{2}(S)\hookrightarrow L^{2}(S)\), by Lemma 3.5 it follows that
$$\phi \bigl(t_{n},\theta_{-t_{n}}\omega,x_{n}( \theta_{-t_{n}}\omega) \bigr)\rightarrow\phi_{0} \mbox{ strongly in } \mathbb{I}(S), $$
which shows that for any given \(\varepsilon>0\), there exists \(N_{1}=N_{1}(\mathbb{K}(\omega),\omega,\varepsilon)\) such that for all \(n\geq N_{1}\),
$$ \bigl\| \phi \bigl(t_{n},\theta_{-t_{n}} \omega,x_{n}(\theta_{-t_{n}}\omega) \bigr)-\phi_{0}\bigr\| ^{2}_{\mathbb{I}(S)}\leq\varepsilon. $$
(3.56)
On the other hand, by Lemma 3.6, there exist \(T_{1}=T_{1}(\mathbb{K}(\omega),\omega,\varepsilon)\) and \(N_{2}=N_{2}(\omega,\varepsilon)\) (large enough) such that \(t_{n}\geq T_{1}\) for every \(n\geq N_{2}\), we have
$$ \int_{|x|>m_{1}}\bigl|\phi \bigl(t_{n}, \theta_{-t_{n}}\omega,x_{n}(\theta_{-t_{n}}\omega ) \bigr) (x)\bigr|^{2}\,dx\leq\varepsilon. $$
(3.57)
Since \(\delta\in\mathbb{H}\cap\mathbb{W}^{2,4}\), there exists \(m_{2}=m_{2}(\varepsilon)\) such that
$$ \int_{|x|>m_{2}}\bigl|\phi_{0}(x)\bigr|^{2}\,dx \leq\varepsilon. $$
(3.58)
Let \(m=\max\{m_{1},m_{2}\}\) and \(N=\max\{N_{1},N_{2}\}\). By (3.56)-(3.58) we find that for all \(n\geq N\),
$$\begin{aligned} &\bigl\| \phi \bigl(t_{n},\theta_{-t_{n}}\omega,x_{n}( \theta_{-t_{n}}\omega) \bigr) (x)-\phi _{0}(x) \bigr\| ^{2}_{\mathbb{I}} \\ &\quad\leq\int_{|x|\leq m}\bigl|\phi \bigl(t_{n}, \theta_{-t_{n}}\omega,x_{n}(\theta _{-t_{n}}\omega) \bigr) (x)-\phi_{0}(x)\bigr|^{2}\,dx \\ &\qquad{}+\int_{|x|>m}\bigl|\phi \bigl(t_{n}, \theta_{-t_{n}}\omega,x_{n}(\theta_{-t_{n}}\omega ) \bigr) (x)-\phi_{0}(x)\bigr|^{2}\,dx \\ &\quad\leq3\varepsilon. \end{aligned}$$
This completes the proof. □