This paper is concerned with the pullback attractors for the Kirchhoff type BBM equations defined on unbounded domains. Sobolev embeddings are invalid on unbounded domains. We obtain the pullback asymptotic compactness of such non-autonomous BBM equations by using the method of uniform tail-estimates.
Consider the following Kirchhoff type BBM equation in unbounded domains: \[\begin{aligned} \label{eq1.1} u_{t}-\Delta u_{t}-\nu \Delta u+\nabla \cdot \overrightarrow{F} (u) -M(\|\nabla u\|^{2})\Delta u =g(x,t), \end{aligned} \tag{1}\] with the initial value conditions \[\begin{aligned} \label{eq1.2} u(x,\tau)=u_\tau(x), u(x,t)=0. \end{aligned} \tag{2}\]
Assume \(M(\cdot)\in C^1(\mathbb{R}),\) \(M^{\prime}(s)\geq0\) \[\begin{aligned} \label{eq1.3} 0\leq M(s)\leq C, \end{aligned} \tag{3}\] where \(x\in\mathcal{O}\), and \(\nu\) is positive constant, \(\mathcal{O}_{0}\) is a bounded open subset in \(\mathbb{R}^2\), \(\mathcal{O}=\mathcal{O}_{0}\times\mathbb{R}\) is a unbounded channel, \(g\in L_{\mathrm{loc}}^{2}(\mathbb{R},L^{2}(\mathcal{O}))\) is a deterministic time-dependent external forcing term, \(\overrightarrow{F}\) is a nonlinear vector function give by \[\begin{aligned} \label{eq1.4} \overrightarrow {F}(s)=\begin{pmatrix}F_1(s),F_2(s),F_3(s)\end{pmatrix}, s\in\mathbb{R}. \end{aligned} \tag{4}\]
Benjamin, Bona and Mahony proposed the BBM equation in [3], which describes the mathematical model of long waves propagation through nonlinear dispersion and dissipation effects. Then many scholars have studied the asymptotic behavior of BBM equations in [2, 14, 15, 16] on bounded domains and in [6, 13, 17] on unbounded domains. The existence and uniqueness of solutions to the BBM equation have been studied in [1, 5, 9, 10]. In recent years, Chen studied the existence and uniqueness of pullback random attractors of BBM equations driven by additive noise, white noise or colored noise, see [7, 6, 17, 12]. Other equations with Kirchhoff terms have been studied by many scholars. In [18], the authors showed the asymptotic behavior for kirchhoff type stochastic plate equations on unbounded domains. In [19], the authors showed the random attractors of Kirchhoff-type reaction-diffusion equations without uniqueness driven by nonlinear colored noise. With the help of reference [14, 4, 11]. We studied the pullback attractors for the Kirchhoff type BBM equations defined on unbounded channels. Due to the Sobolev embedding theorem is noncompact on unbounded domain, to conquer this difficulty we use the uniform tail estimation of the solution.
The structure of our article is as follows. In Section 2, we review some results of the pullback attractor. In Section 3, we discuss the uniform tail-estimates of solutions. In Section 4, we prove the existence of the pullback attractor of the Kirchhoff BBM equation.
In this paper, we use \(||\cdot||\) and \((\cdot, \cdot)\) to denote the norm and the inner product of \(L^2(\mathcal{O})\), respectively. If \(u\in{H_{0}^{1}(\mathcal{O})}\), we define \(\|u\|_{H_0^1(\mathcal{O})}=\|\nabla u\|\) and \(\|u\|_{H^1(\mathcal{O})}=\left(\|u\|^2+\|\nabla u\|^2\right)^{\frac12}.\) For easy of writing, each \(C\) used in this article represents a different positive constant.
In this paper, we will use the following Poincar\(\acute{e}\) inequality : \[\begin{aligned} \label{eq2.1} \|\nabla u\|^2\geq\lambda\|u\|^2,\quad\forall u\in H_0^1(\mathcal{O}), \end{aligned} \tag{5}\] where, \[\begin{aligned} \label{eq2.2} \gamma=\min\Big\{\nu,\frac{1}{4}\nu\lambda\Big\}. \end{aligned} \tag{6}\]
We define that \({G_i}\left(i=1,2,3\right), {F_i}\left(i=1,2,3\right)\) meet the following conditions: \[\begin{aligned} \label{eq2.3} G_{i}(u)=\int_{0}^{u}F_{i}(t)dt, \overrightarrow{G}(u)=\big(G_{1}(u),G_{2}(u),G_{3}(u)\big),\quad u\in\mathbb{R}, \end{aligned} \tag{7}\] \[\begin{aligned} \label{eq2.4} \quad|F_i(u)|\leq |u|+C|u|^2,|F_i'(u)|\leq1+C|u|,|G_i(u)|\leq C|u|^2+C|u|^3,\quad u\in\mathbb{R} \end{aligned} \tag{8}\] where \(t, u\in\mathbb{R}\), \(0\leq p<1\).
By [14], we get that for \(\beta\geqslant0\) and \(0\leqslant\alpha<\frac\gamma2\) such that \[\begin{aligned} \label{eq2.5} \|g(t)\|^2\leqslant\beta e^{\alpha|t|} \end{aligned} \tag{9}\] and \[\begin{aligned} \label{eq2.6} \int_{-\infty}^{\tau}e^{\gamma r}\|g(r)\|^2dr<\infty\quad\text{for all }\tau\in\mathbb{R}, \end{aligned} \tag{10}\]
\[\begin{aligned} \label{eq2.7} \int_{-\infty}^\tau\left(\int_{-\infty}^re^{\gamma s/2}\|g(s)\|^2ds\right)^2dr<\infty\quad\text{for all }\tau\in\mathbb{R}. \end{aligned} \tag{11}\]
Furthermore \[\begin{aligned} \label{eq2.8} \lim_{k\to\infty}\int_{-\infty}^{\tau}\int_{|x_3|\geqslant k}e^{\gamma r}|g(x,r)|^2dxdr=0\quad\text{for all }\tau\in\mathbb{R}. \end{aligned} \tag{12}\]
Let \(P=\mathbb{R}\) and define \(\Phi:\mathbb{R}^+\times P\times H_0^1(\mathcal{O})\rightarrow H_0^1(\mathcal{O})\) by \[\begin{aligned} \label{eq2.9} \Phi(t,\tau,u_\tau)&=u(t+\tau,\tau,u_\tau), \end{aligned} \tag{13}\] and \[\begin{aligned} \label{eq2.10} \Phi(t+s,\tau,u_\tau)=\Phi(t,s+\tau,\Phi(s,\tau,u_\tau)),&\quad t,s\geqslant0,\tau\in\mathbb{R},u_\tau\in H_0^1(\mathcal{O}). \end{aligned} \tag{14}\]
Set \(\mathcal{R}_\gamma=\{r:\mathbb{R}\to(0,\infty):\lim_{s\to-\infty}e^{\gamma s}r^{2}(s)=0\}\) and \[\begin{aligned} \mathcal{B}_\gamma=\{B=\{B(s)\}_{s\in\mathbb{R}}|B(s)\subset\overline{D}(0,r(s))\quad \text{for some} \ r\in\mathcal{R}_\gamma, \end{aligned} \tag{15}\] where \(\overline D(0,r(s))\) denotes the closed ball in \(H_0^1(\mathcal{O})\).
Set \(P\) be a non-empty set and \(X\) a metric space with distance \(d(\cdot,\cdot).\)
Definition 2.1. A family of mappings \(\left \{ \theta _t\right \} _{t\in \mathbb{R} }\) from \(P\) to itself is called a family of shift operators on \(P\), if \(\left \{ \theta _t\right \} _{t\in \mathbb{R} }\) satisfies the following group properties:
(i) \(\theta _0( p) = p\) for all \(p\in P\),
(ii) \(\theta _{t+ \tau} ( p) = \theta _t( \theta _\tau ( p) )\) for all \(p\in P\) and \(t\), \(\tau \in \mathbb{R}.\)
Definition 2.2. Let \({\left \{ \theta _t\right \}} _{t\in \mathbb{R} }\) be a family of shift operators on \(P\). Then a continuous \(\theta\)-cocycle \(\Phi\) on \(X\) is a mapping \(\Phi:\mathbb{R} ^+ \times P\times X\to X\) satisfying, for all \(p\in P\) and \(t,\tau \in \mathbb{R}\),
(i) \(\Phi( 0, q, \cdot )\) is the identity on \(X\);
(ii) \(\Phi(t+\tau,p,\cdot)=\Phi(t,\theta_\tau(p),\Phi(\tau,p,\cdot));\)
(iii) \(\Phi ( t, p, \cdot ) : X\to X\) is continuous.
Suppose \(\mathcal{B}\) is a nonempty class of parameterized sets \(B=\{B(p)\}_{p\in P}, B(P)\subset X\) for every \(p\in P.\)
Definition 2.3. It is said \(D=\{D(p)\}_{p\in B}\in\mathcal{B}\) is pullback \(\mathcal{B}\)-absorbing for \(\Phi\) in \(\mathcal{B}\) if for every \(p\in P\) and \(B\in \mathcal{B}\), there exists \(T(p,B)>0\) such that \[ \Phi(t,\theta_{-t}(p),B(\theta_{-t}(p)))\subset D(p),\quad\forall t\geqslant T(p,B). \tag{16}\]
Definition 2.4. The cocycle \(\Phi\) is said to be pullback \(\mathcal{B}\)-asymptotically compact in \(X\), if for every \(p\in P\), \(\Phi ( t_n, \theta _{- t_n}( p)\) , \({x_n)}\) has a convergent subsequence in \(X\) whenever \(t_n\to + \infty\) and \(x_n\in B( \theta _{- t_n}( p) )\) with \(\{B(p)\}_{p\in P}\in\mathcal{B}.\)
Definition 2.5. A family \(C=\{C(p)\}_{p\in P}\in\mathcal{B}\) is said to be pullback \(\mathcal{B}\)-attracting if \[ \lim_{t\to\infty}\operatorname{dist}(\Phi(t,\theta_{-t}(p),B(\theta_{-t}(p)),C(p))=0 \quad for all p\in P,B\in\mathcal{B}, \tag{17}\] where \(\operatorname{dist}(X,Y)=\sup_{x\in X}\inf_{y\in Y}d(x,y)\) is the Hausdorff semi-distance between \(X\) and \(Y\).
Definition 2.6. It is said that \(A= \{ A( p) \} _{p\in P}\in \mathcal{B}\) is a pullback \(\mathcal{B}\)-attractor if it satisfies
(1) \(A( p)\) is compact in \(X\) for any \(p\in P;\)
(2) \(A\) is pullback \(\mathcal{B}\)-attracting;
(3) \(A\) is invariant, that is, \(\Phi( t, p, A( p) ) = A( \theta _t( p) )\) for all \(( t, p) \in \mathbb{R}^{+} \times P.\)
Theorem 2.7. Let \(\Phi\) be a continuous \(\theta\)-cocycle on \(X\) and there exists a family of pullback \(\mathcal{B}\)-absorbing sets \(\{D(p)\}_{p\in P}\in\mathcal{B}\). If \(\Phi\) is a pullback \(\mathcal{B}\)-asymptotically compact in \(X\), then \(\Phi\) has a unique pullback \(\mathcal{B}\)-attractor \(\left\{ A( p) \right\} _{p\in P}\in \mathcal{B}\) defined by \[\begin{aligned} \label{eq2.13} A(p)=\bigcap_{\tau\geqslant0}\overline{\bigcup_{t\geqslant\tau}\Phi(t,\theta_{-t}(p),D(\theta_{-t}(p)))}. \end{aligned} \tag{18}\]
Lemma 3.1. For every \(\tau\in\mathbb{R},B=\{B(s)\}_{s\in\mathbb{R}}\in\mathcal{B}_{\gamma}\) and \(u_{0}(\tau-t)\in B(\tau-t)\), there exists \(T=T(\tau,B)>0\) such that, for all \(t\geqslant T\) \[\begin{aligned} \label{eq3.1} \|u(\tau,\tau-t,u_0(\tau-t))\|_{H_0^1(\mathcal{O})}^2&+\int_{\tau-t}^{s}e^{\gamma r}(M(\|\nabla u\|^2)\|\nabla u\|_{H_0^1(\mathcal{O})}^2dr \nonumber\\ &\leqslant\frac{2}{\nu\lambda}e^{-\gamma\tau}\int_{-\infty}^{\tau}e^{\gamma r}\|g(r)\|^2dr, \end{aligned} \tag{19}\] \[\begin{aligned} \label{eq3.2} \int_{\tau-t}^{\tau}e^{\gamma r}\|u(r,\tau-t,u_{0}(\tau-t))\|_{H_{0}^{1}(\mathcal{O})}^{2}dr\leqslant\frac{2}{\nu\lambda\gamma}\int_{-\infty}^{\tau}e^{\gamma r}\|g(r)\|^{2}dr, \end{aligned} \tag{20}\] and \[\begin{aligned} \label{eq3.3} \|u(s,\tau-t,u_0(\tau-t))\|_{H_0^1(\mathcal{O})}^2&+\int_{\tau-t}^{s}e^{\gamma r/2}(M(\|\nabla u\|^2)\|\nabla u\|_{H_0^1(\mathcal{O})}^2dr \nonumber\\ &\leqslant\frac{2}{\nu\lambda}e^{-\gamma s/2}\int_{-\infty}^{s}e^{\gamma r/2}\|g(r)\|^2dr,s\in[\tau-t,\tau]. \end{aligned} \tag{21}\]
Proof. Suppose \(\tau\in\mathbb{R},t\geqslant0,B=\{B(s)\}_{s\in\mathbb{R}}\in\mathcal{B}_{\gamma},u_{0}(\tau-t)\in B(\tau-t)\), and define \(u(r)=u(r,\tau-t,u_{0}(\tau-t))\) for \(r\geqslant\tau-t\). Then by (1) \(u(r)\) in \(L^2(\mathcal{O})\), we get
\[\begin{aligned} \label{eq3.4} \frac{1}{2}\frac{d}{dr}(\|u\|^2+\|\nabla u\|^2)+\nu\|\nabla u\|^2+M(\|\nabla u\|^2)\|\nabla u\| +\int_{\mathcal{O}}(\nabla\cdot\overrightarrow{F}(u))udx=(g,u). \end{aligned} \tag{22}\]
We have \[\begin{aligned} \label{eq3.5} \int_{\mathcal{O}}(\nabla\cdot\overrightarrow{F}(u))udx=-\int_{\mathcal{O}}\overrightarrow{F}(u)\cdot\nabla udx=-\int_{\mathcal{O}}\nabla\cdot\overrightarrow{G}(u)dx=0. \end{aligned} \tag{23}\]
By Young’s inequality, we obtain \[\begin{aligned} \label{eq3.6} \frac{1}{2}\frac{d}{dr}(\|u\|^2+\|\nabla u\|^2)+\nu\|\nabla u\|^2+M(\|\nabla u\|^2)\|\nabla u\| &\leqslant\frac{\nu\lambda}{2}\|u\|^2+\frac{1}{2\nu\lambda}\|g\|^2\notag\\ &\leqslant\frac{\nu}{2}\|\nabla u\|^2+\frac{1}{2\nu\lambda}\|g\|^2, \end{aligned} \tag{24}\]
Set \(s\in[\tau-t,\tau]\). By Gronwall lemma and integrating it over \((\tau-t,s)\), we get \[\begin{aligned} \label{eq3.7} &\|u(s,\tau-t,u_{0}(\tau-t))\|_{H_{0}^{1}(\mathcal{O})}^{2}+\int_{\tau-t}^{s}e^{\gamma r}(M(\|\nabla u\|^2)\|\nabla u\|_{H_0^1(\mathcal{O})}^2dr\nonumber\\ &\quad+\gamma\int_{\tau-t}^{s}e^{\gamma r}\|u(r,\tau-t,u_{0}(\tau-t))\|_{H_{0}^{1}(\mathcal{O})}^{2}dr \nonumber\\ &\leqslant e^{\gamma(\tau-t)}\|u_{0}(\tau-t)\|_{H_{0}^{1}(\mathcal{O})}^{2}+\frac{1}{\nu\lambda}\int_{-\infty}^{s}e^{\gamma r}\|g(r)\|^{2}dr. \end{aligned} \tag{25}\]
Note that \(u_0(\tau-t)\in B(\tau-t)\) and \(B=\{B(s)\}_{s\in\mathbb{R}}\in\mathcal{B}_\gamma\), \(T=T(s,B)>0\) , \[\begin{aligned} \label{eq3.8} e^{\gamma(\tau-t)}\|u_0(\tau-t)\|_{H_0^1(\mathcal{O})}^2\leqslant\frac{1}{\nu\lambda}\int_{-\infty}^se^{\gamma r}\|g(r)\|^2dr,\quad t\geqslant T. \end{aligned} \tag{26}\]
By (25), we get \[\begin{aligned} \label{eq3.9} \|u(s,\tau-t,u_0(\tau-t))\|_{H_0^1(\mathcal{O})}^2&+\int_{\tau-t}^{s}e^{\gamma r}(M(\|\nabla u\|^2)\|\nabla u\|_{H_0^1(\mathcal{O})}^2dr \nonumber\\ &\leqslant\frac{2}{\nu\lambda}e^{-\gamma s}\int_{-\infty}^se^{\gamma r}\|g(r)\|^2dr. \end{aligned} \tag{27}\]
By (27), we know that \[\begin{aligned} \label{eq3.10} \|u(s,\tau-t,u_0(\tau-t))\|_{H_0^1(\mathcal{O})}^2&+\int_{\tau-t}^{s}e^{\gamma r}(M(\|\nabla u\|^2)\|\nabla u\|_{H_0^1(\mathcal{O})}^2dr \nonumber\\ &\leqslant\frac{2}{\nu\lambda}e^{-\gamma s/2}\int_{-\infty}^se^{\gamma r/2}\|g(r)\|^2dr,\quad s\in[\tau-t,\tau]. \end{aligned} \tag{28}\] This completes the proof of Lemma 3.1. ◻
Lemma 3.2. For every \(\tau\in\mathbb{R},B=\{B(s)\}_{s\in\mathbb{R}}\in\mathcal{B}_\gamma\) and \(u_0(\tau-t)\in B(\tau-t)\), there exists \(T=T(\tau,B)>0\) such that, for all \(t\geqslant T\)
\[\begin{aligned} \label{eq3.11} \int_{\tau-t}^\tau e^{\gamma r}\|u_r\|_{H_0^1(\mathcal{O})}^2dr\leqslant C\int_{-\infty}^\tau e^{\gamma r}\|g(r)\|^2dr+C\int_{-\infty}^\tau\left(\int_{-\infty}^re^{\gamma s/2}\|g(s)\|^2ds\right)^2dr. \end{aligned} \tag{29}\]
Proof. Taking the inner product of (1) with \(u_r(r)\) in \(L^2(\mathcal{O})\) and using Young’s inequality, we get \[\begin{aligned} \label{eq3.12} \|u_{r}\|^{2}+\|\nabla u_{r}\|^{2}&=-\nu(\nabla u,\nabla u_r)-M(\|\nabla u\|^2)(\nabla u,\nabla u_r)-\int_{\mathcal{O}}(\nabla\cdot\overrightarrow{F}(u))u_rdx+(g,u_r)\nonumber\\ &\leqslant\frac{1}{4}\|\nabla u_{r}\|^{2}+\nu^{2}\|\nabla u\|^{2}+\frac{1}{4}\|\nabla u_{r}\|^{2}+M(\|\nabla u\|^2)^{2}\|\nabla u\|^{2}\nonumber\\ &\quad-\int_{\mathcal{O}}(\nabla\cdot\overrightarrow{F}(u))u_{r}dx+\frac{1}{2}\|g\|^{2}+\frac{1}{2}\|u_{r}\|^{2}. \end{aligned} \tag{30}\]
Using (8), we obtain \[\begin{aligned} \label{eq3.13} -\int_{\mathcal{O}}(\nabla\cdot\overrightarrow{F}(u))u_{r}dx&=\int_{\mathcal{O}}\overrightarrow{F}(u)\cdot\nabla u_{r}dx\nonumber\\ &\leqslant C\int_{\mathcal{O}}|u||\nabla u_{r}|dx+C\int_{\mathcal{O}}|u|^{2}|\nabla u_{r}|dx\nonumber\\ &\leqslant\frac{1}{2}\|\nabla u_r\|^2+C(\|u\|^2+\|u\|_4^4)\nonumber\\ &\leqslant\frac{1}{2}\|\nabla u_r\|^2+C(\|u\|_{H_0^1(\mathcal{O})}^2+\|u\|_{H_0^1(\mathcal{O})}^4). \end{aligned} \tag{31}\]
Therefore, we have \[\begin{aligned} \label{eq3.14} \|u_r\|^2+\|\nabla u_r\|^2\leqslant C(\|u\|_{H_0^1(\mathcal{O})}^2+\|g\|^2+\|u\|_{H_0^1(\mathcal{O})}^4). \end{aligned} \tag{32}\]
Multiplying this by \(e^{\gamma r}\), integrating it over \((\tau-t,\tau)\), we have \[\begin{aligned} \label{eq3.15} \int_{\tau-t}^{\tau}e^{\gamma r}(\|u_{r}\|^{2}+\|\nabla u_{r}\|^{2})dr\leqslant C\int_{-\infty}^{\tau}e^{\gamma r}\|g(r)\|^{2}dr+C\int_{\tau-t}^{\tau}e^{\gamma r}\|u(r)\|_{H_{0}^{1}(\mathcal{O})}^{4}dr. \end{aligned} \tag{33}\]
By (21), we get \[\int_{\tau-t}^{\tau}e^{\gamma r}\|u(r)\|_{H_{0}^{1}(\mathcal{O})}^{4}dr\leqslant C\int_{\tau\boldsymbol{-}t}^{\tau}e^{\gamma r}\left(e^{\boldsymbol{-}\gamma r/2}\int_{-\infty}^{r}e^{\gamma s/2}\|g(s)\|^{2}ds\right)^{2}dr\] \[\leqslant C\int_{-\infty}^{\tau}\left(\int_{-\infty}^{r}e^{\gamma s/2}\|g(s)\|^{2}ds\right)^{2}dr. \tag{34}\] ◻
Lemma 3.3. For every \(\epsilon>0,\tau\in\mathbb{R}\) and \(B=\{B(s)\}_{s\in\mathbb{R}}\in\mathcal{B}_{\delta}\), there exist \(T=T(\tau,B,\epsilon)>0\) and \(K=K(\tau,\epsilon)>1\) such that for all \(t\geqslant T\) and \(k\geqslant K\), \[\begin{aligned} \label{eq3.17} \int_{|x_3|\geqslant k}(|u(x,\tau,\tau-t,u_0(\tau-t))|^2+|\nabla u(x,\tau,\tau-t,u_0(\tau-t))|^2)dx\leqslant\epsilon, \end{aligned} \tag{35}\] where \(u_0( \tau – t) \in B( \tau – t)\).
Proof. Let \(k\geqslant1\) and take a smooth function \(\varphi\) such that \(0\leqslant\varphi\leqslant1\) for all \(s\in\mathbb{R}^+\) and
\[\begin{aligned} \label{eq3.18} \left.\varphi(s)=\left\{\begin{array}{ll}0,&\quad0\leqslant s\leqslant1,\\1,&\quad s\geqslant2.\end{array}\right.\right. \end{aligned} \tag{36}\]
Let \(|\ \varphi^{\prime}(s)|\leqslant C\) for all \(s\in\mathbb{R}^+\). Taking the inner product of (1) by \(\varphi^2(\frac{x_3^2}{k^2})u,\) we observe that
\[\begin{aligned} \label{eq3.19} \frac{d}{dr}&\int_{\mathcal{O}}\varphi^{2}\left(\frac{x_{3}^{2}}{k^{2}}\right)(|u|^{2}+|\nabla u|^{2})dx+2\nu\int_{\mathcal{O}}\varphi^{2}\left(\frac{x_{3}^{2}}{k^{2}}\right)|\nabla u|^{2}dx\nonumber\\ &\quad+2M(\|\nabla u\|^2)\int_{\mathcal{O}}\varphi^{2}\left(\frac{x_{3}^{2}}{k^{2}}\right)|\nabla u|^{2}dx\nonumber\\ &=-2\int_{\mathcal{O}}\left(\nabla\varphi^{2}\left(\frac{x_{3}^{2}}{k^{2}}\right)\cdot\nabla u_{r}\right)udx-2\nu\int_{\mathcal{O}}\left(\nabla\varphi^{2}\left(\frac{x_{3}^{2}}{k^{2}}\right)\cdot\nabla u\right)udx\nonumber\\ &\quad-2M(\|\nabla u\|^2)\int_{\mathcal{O}}\left(\nabla\varphi^{2}\left(\frac{x_{3}^{2}}{k^{2}}\right)\cdot\nabla u\right)udx\nonumber\\ &\quad-2\int_{\mathcal{O}}(\nabla\cdot\overrightarrow{F}(u))\varphi^{2}\left(\frac{x_{3}^{2}}{k^{2}}\right)udx+2\int_{\mathcal{O}}g\varphi^{2}\left(\frac{x_{3}^{2}}{k^{2}}\right)udx\nonumber\\ &:=J_1+J_2+J_3+J_4+J_5. \end{aligned} \tag{37}\]
Then, we will handle each term in (37) we have
\[\begin{aligned} \label{eq3.20} J_{1}&\leqslant2\int_{\mathcal{O}}|2\varphi\varphi^{\prime}\left(\frac{x_{3}^{2}}{k^{2}}\right)|\frac{2|x_{3}|}{k^{2}}|\nabla u_{r}||u|dx\nonumber\\ &\leqslant C\int_{k\leqslant|x_{3}|\leqslant\sqrt{2}k}\frac{2|x_{3}|}{k^{2}}|\nabla u_{r}||u|dx\nonumber\\ &\leqslant\frac{C}{k}\int_{k\leqslant|x_3|\leqslant\sqrt{2}k}|\nabla u_r||u|dx\nonumber\\ &\leqslant\frac{C}{k}(\|\nabla u_r\|^2+\|u\|^2). \end{aligned} \tag{38}\]
Similarly, \[\begin{aligned} \label{eq3.21} J_{2}\leqslant\frac{C}{k}(\|\nabla u\|^{2}+\|u\|^{2}). \end{aligned} \tag{39}\]
\[\begin{aligned} \label{eq3.22} J_{3}\leqslant\frac{C}{k}(\|\nabla u\|^{2}+\|u\|^{2}). \end{aligned} \tag{40}\]
We get \[\begin{aligned} \label{eq3.23} J_{4}&=2\int_{\mathcal{O}}(\overrightarrow{F}(u)\cdot\nabla u)\varphi^{2}\left(\frac{x_{3}^{2}}{k^{2}}\right)dx+2\int_{\mathcal{O}}\left(\overrightarrow{F}(u)\cdot\nabla\varphi^{2}\left(\frac{x_{3}^{2}}{k^{2}}\right)\right)udx. \end{aligned} \tag{41}\]
By Green formula and \(\overrightarrow{G}(u)=\overrightarrow{0}\), we obtain \[\begin{aligned} \label{eq3.24} 2\int_{\mathcal{O}}(\overrightarrow{F}(u)\cdot\nabla u)\varphi^{2}\left(\frac{x_{3}^{2}}{k^{2}}\right)dx&=2\int_{\mathcal{O}}(\nabla\cdot\overrightarrow{G}(u))\varphi^{2}\left(\frac{x_{3}^{2}}{k^{2}}\right)dx\nonumber\\ &=-2\int_{\mathcal{O}}\overrightarrow{G}(u)\cdot\nabla\varphi^{2}\left(\frac{x_{3}^{2}}{k^{2}}\right)dx. \end{aligned} \tag{42}\]
Using (42) and (7), (8), we get \[\begin{aligned} \label{eq3.25} J_{4}&\leqslant2\int_{\mathcal{O}}|\overrightarrow{G}(u)||2\varphi\varphi^{\prime}\left(\frac{x_{3}^{2}}{k^{2}}\right)|\frac{2|x_{3}|}{k^{2}}dx+2\int_{\mathcal{O}}|\overrightarrow{F}(u)||2\varphi\varphi^{\prime}\left(\frac{x_{3}^{2}}{k^{2}}\right)|\frac{2|x_{3}|}{k^{2}}|u|dx\nonumber\\ &\leqslant\frac{C}{k}\int_{k\leqslant|x_{3}|\leqslant\sqrt{2}k}|\overrightarrow{G}(u)|dx+\frac{c}{k}\int_{k\leqslant|x_{3}|\leqslant\sqrt{2}k}|\overrightarrow{F}||u|dx\nonumber\\ &\leqslant\frac{C}{k}\int_{\mathcal{O}}(|u|^2+|u|^3)dx\leqslant\frac{C}{k}(\|u\|^2+\|u\|_3^3)\nonumber\\ &\leqslant\frac{C}{k}(\|u\|^2+\|u\|_{H_0^1(\mathcal{O})}^3)\leqslant\frac{C}{k}(1+\|u\|^2+\|u\|_{H_0^1(\mathcal{O})}^4). \end{aligned} \tag{43}\]
Using Young’s inequality, we have
\[\begin{aligned} \label{eq3.26} J_{5}&=2\int_{|x_{3}|\geqslant k}g\varphi^{2}\left(\frac{x_{3}^{2}}{k^{2}}\right)udx\nonumber\\ &\leqslant\frac{\nu\lambda}{4}\int_{|x_{3}|\geqslant k}\varphi^{2}\left(\frac{x_{3}^{2}}{k^{2}}\right)u^{2}dx+\frac{4}{\nu\lambda}\int_{|x_{3}|\geqslant k}\varphi^{2}\left(\frac{x_{3}^{2}}{k^{2}}\right)|g|^{2}dx\nonumber\\ &\leqslant\frac{\nu\lambda}{4}\int_{\mathcal{O}}\varphi^{2}\left(\frac{x_{3}^{2}}{k^{2}}\right)u^{2}dx+\frac{4}{\nu\lambda}\int_{|x_{3}|\geqslant k}|g|^{2}dx. \end{aligned} \tag{44}\]
By (38)-(44) , we get
\[\begin{aligned} \label{eq3.27} &\frac{d}{dr}\int_{\mathcal{O}}\varphi^{2}\left(\frac{x_{3}^{2}}{k^{2}}\right)(|u|^{2}+|\nabla u|^{2})dx+2\nu\int_{\mathcal{O}}\varphi^{2}\left(\frac{x_{3}^{2}}{k^{2}}\right)|\nabla u|^{2}dx\nonumber\\ &\leqslant\frac{C}{k}(1+\|u\|_{H_{0}^{1}(\mathcal{O})}^{2}+\|\nabla u_{r}\|^{2}+\|u\|_{H_{0}^{1}(\mathcal{O})}^{4})\nonumber\\ &+\frac{\nu\lambda}{4}\int_{\mathcal{O}}\varphi^{2}\left(\frac{x_{3}^{2}}{k^{2}}\right)u^{2}dx+\frac{4}{\nu\lambda}\int_{|x_{3}|\geqslant k}|g|^{2}dx. \end{aligned} \tag{45}\]
By [14], \[\begin{aligned} \label{eq3.28} 2\nu\int_{\mathcal{O}}\varphi^{2}\left(\frac{x_{3}^{2}}{k^{2}}\right)|\nabla u|^{2}dx\geqslant\nu\int_{\mathcal{O}}\varphi^{2}\left(\frac{x_{3}^{2}}{k^{2}}\right)|\nabla u|^{2}dx+\frac{\nu\lambda}{2}\int_{\mathcal{O}}\varphi^{2}\left(\frac{x_{3}^{2}}{k^{2}}\right)u^{2}dx-\frac{C}{k^{2}}\|u\|^{2}, \end{aligned} \tag{46}\]
Similarly, \[\begin{aligned} \label{eq3.29} -2\int_{\mathcal{O}}M(\|\nabla u\|^2)\varphi^{2}\left(\frac{x_{3}^{2}}{k^{2}}\right)|\nabla u(t)|^{2}dx&\leq-M(\|\nabla u\|^2)\int_{\mathcal{O}}\varphi^{2}\left(\frac{x_{3}^{2}}{k^{2}}\right)|\nabla u|^2dx\nonumber\\ &\quad -\frac{M(\|\nabla u\|^2)\lambda}2\int_{\mathcal{O}}\varphi^{2}\left(\frac{x_{3}^{2}}{k^{2}}\right)|u|^2dx+\frac C{k^2}\|u\|^2\nonumber\\ &\leq\frac C{k}\|u\|^2, \end{aligned} \tag{47}\] we get \[\begin{aligned} \label{eq3.30} &\frac{d}{dr}\int_{\mathcal{O}}\varphi^{2}\left(\frac{x_{3}^{2}}{k^{2}}\right)(|u|^{2}+|\nabla u|^{2})dx+\nu\int_{\mathcal{O}}\varphi^{2}\left(\frac{x_{3}^{2}}{k^{2}}\right)|\nabla u|^{2}dx+\frac{\nu\lambda}{4}\int_{\mathcal{O}}\varphi^{2}\left(\frac{x_{3}^{2}}{k^{2}}\right)u^{2}dx\nonumber\\ &\leqslant\frac{4}{\nu\lambda}\int_{|x_{3}|\geqslant k}|g|^{2}dx+\frac{C}{k}(1+\|u\|_{H_{0}^{1}(\mathcal{O})}^{2}+\|\nabla u_{r}\|^{2}+\|u\|_{H_{0}^{1}(\mathcal{O})}^{4}). \end{aligned} \tag{48}\]
Finally, Using Gronwall lemma, integrating it over \((\tau-t,\tau)\) and Lemma 3.1, Lemma 3.2 and (17), we have
\[\begin{aligned} \label{eq3.31} e^{\gamma\tau}&\int_{\mathcal{O}}\varphi^{2}\left(\frac{x_{3}^{2}}{k^{2}}\right)(|u(x,\tau,\tau-t,u_{0}(\tau-t))|^{2}+|\nabla u(x,\tau,\tau-t,u_{0}(\tau-t))|^{2})dx\nonumber\\ &\leqslant e^{\gamma(\tau\boldsymbol{-}t)}\int_{\mathcal{O}}\varphi^{2}\left(\frac{x_{3}^{2}}{k^{2}}\right)(|u_{0}|^{2}+|\nabla u_{0}|^{2})dx+\frac{4}{\nu\lambda}\int_{\tau\boldsymbol{-}t}^{\tau}e^{\gamma r}\int_{|x_{3}|\geqslant k}|g(x,r)|^{2}dxdr\nonumber\\ &\quad+\frac{C}{k}\int_{\tau-t}^{\tau}e^{\gamma r}(1+\|u\|_{H_{0}^{1}(\mathcal{O})}^{2}+\|\nabla u_{r}\|^{2}+\|u(r)\|_{H_{0}^{1}(\mathcal{O})}^{4})dr\nonumber\\ &\leqslant e^{\gamma(\tau\boldsymbol{-}t)}\int_{\mathcal{O}}\varphi^{2}\left(\frac{x_{3}^{2}}{k^{2}}\right)(|u_{0}|^{2}+|\nabla u_{0}|^{2})dx+\frac{4}{\nu\lambda}\int_{\tau\boldsymbol{-}t}^{\tau}e^{\gamma r}\int_{|x_{3}|\geqslant k}|g(x,r)|^{2}dxdr\nonumber\\ &\quad+\frac{C}{k}\int_{-\infty}^{\tau}e^{\gamma r}\|g(r)\|^{2}dr+\frac{C}{k}(e^{\gamma\tau}-e^{\gamma(\tau-t)})+\frac{C}{k}\int_{-\infty}^{\tau}\left(\int_{-\infty}^{r}e^{\gamma s/2}\|g(s)\|^{2}ds\right)^{2}dr, \end{aligned} \tag{49}\]
and \[\begin{aligned} \label{eq3.32} &\int_{|x_{3}|\geqslant\sqrt{2}k}(|u(x,\tau,\tau-t,u_{0}(\tau-t))|^{2}+|\nabla u(x,\tau,\tau-t,u_{0}(\tau-t))|^{2})dx\nonumber\\ &\leqslant e^{-\gamma\tau}e^{\gamma(\tau-t)}\|u_{0}(\tau-t)\|_{H_{0}^{1}(\mathcal{O})}^{2}+\frac{4}{\nu\lambda}e^{-\gamma\tau}\int_{\tau-t}^{\tau}e^{\gamma r}\int_{|x_{3}|\geqslant k}|g(x,r)|^{2}dxdr\nonumber\\ &\quad+\frac{C}{k}+\frac{C}{k}e^{-\gamma\tau}\int_{-\infty}^\tau e^{\gamma r}\|g(r)\|^2dr+\frac{C}{k}e^{-\gamma\tau}\int_{-\infty}^\tau\left(\int_{-\infty}^re^{\gamma s/2}\|g(s)\|^2ds\right)^2dr. \end{aligned} \tag{50}\]
Since \(u_0(\tau-t)\in B(\tau-t)\in\mathcal{B}_\delta\), for given \(\epsilon>0,\) there exists \(T(\tau,B,\epsilon)>0\) such that
\[\begin{aligned} \label{eq3.33} e^{-\gamma\tau}e^{\gamma(\tau-t)}\|u_{0}(\tau-t)\|_{H_{0}^{1}(\mathcal{O})}^{2}\leqslant\frac{\epsilon}{3},\quad t\geqslant T. \end{aligned} \tag{51}\]
By (10)-(12), we get \[\begin{aligned} \label{eq3.34} \frac{4}{\nu\lambda}e^{-\gamma\tau}\int_{\tau-t}^\tau e^{\gamma r}\int_{|x_3|\geqslant k}|g(x,t)|^2dxdr\leqslant\frac{\epsilon}{3}, \end{aligned} \tag{52}\]
\[\begin{aligned} \label{eq3.35} \frac{C}{k}+\frac{C}{k}e^{-\gamma\tau}\int_{-\infty}^{\tau}e^{\gamma r}\|g(r)\|^2dr+\frac{C}{k}\int_{-\infty}^{\tau}\left(\int_{-\infty}^{r}e^{\gamma rs/2}\|g(s)\|^2ds\right)^2dr\leqslant\frac{\epsilon}{3}. \end{aligned} \tag{53}\] ◻
This completes the proof of Lemma 3.3.
Lemma 3.4. Set \(B\) \(\in\mathcal{B}_\gamma\) and \(u_0(\tau-t)\in B(\tau-t).\) Then, for every \(\epsilon>0\) and \(k\geqslant1\), there exist \(T=T(\tau,B,\epsilon)>0\) and an integer \(N=N(\tau,\epsilon)>0\) such that \[\begin{aligned} \label{eq3.36} \|(I-P_{n})v(\tau,\tau-t,u_{0}(\tau-t))\|_{H_{0}^{1}(\mathcal{O}_{2k})}<\epsilon,\quad for all\ t\geqslant T,n\geqslant N. \end{aligned} \tag{54}\] where \(v_{\tau-t}=\xi\biggl(\frac{x_3^2}{k^2}\biggr)u_{\tau-t}\).
Proof. Set \(v=\mathcal{P}_nv+(I-\mathcal{P}_n)v:=v_1+v_2.\) Applying \(I-\mathcal{P}_n\) to (1) and taking the inner product of the resulting equation with \(v_2\mathrm{~in~}L^2(\mathcal{O}_{2k})\), by [8] we have \[\begin{aligned} \label{eq3.37} &\frac{d}{dt}\|v_2\|_{H^1(\mathcal{O}_{2k})}^2+\nu\|\nabla v_2\|^2+M(\|\nabla u\|^2)\|\nabla v_2\|^2+\frac{1}{4}\nu\lambda_{n+1}\|v_2\|^2\nonumber\\ &\leq C(\lambda_{n+1}^{-1}+\lambda_{n+1}^{-\frac{1}{2}})\Big(1+\|g(t)\|^2+\|u\|_{H^1(\mathcal{O})}^4\Big). \end{aligned} \tag{55}\]
If \(\lambda_n\to\infty\) as \(n\to\infty,\) we know that there exists \(N\in\mathbb{N}\) such that \[\begin{aligned} \label{eq3.38} \lambda_{n+1}\geq\max\{\lambda,1\}\quad\mathrm{for~}n\geq N\quad\mathrm{and}\quad\lim_{n\to\infty}\lambda_{n+1}^{-\frac12}=0. \end{aligned} \tag{56}\]
Thus, by (8) and (55) we know that for \(n\geq N\), \[\begin{aligned} \label{eq3.39} \frac{d}{dt}\|v_2\|_{H^1(\mathcal{O}_{2k})}^2+\gamma\|v_2\|_{H^1(\mathcal{O}_{2k})}^2\leq C\lambda_{n+1}^{-\frac{1}{2}}\Big(1+\|g(t)\|^2+\|u\|_{H^1(\mathcal{O})}^4\Big). \end{aligned} \tag{57}\]
Using the Gronwall lemma to (57), and integrating over \((\tau-t,\tau)\) , and \(t\geq0\), we have \[\begin{aligned} \label{eq3.40} \|v_{2}(\tau,\tau-t,v_{2,\tau-t})\|_{H^{1}(\mathcal{O}_{2k})}^{2} &\leq e^{-\gamma t}\|(I-\mathcal{P}_{n})(\xi u_{\tau-t})\|_{H^{1}(\mathcal{O}_{2k})}^{2}\nonumber\\ &\quad+C\lambda_{n+1}^{-\frac{1}{2}}\int_{\tau-t}^{\tau}e^{\gamma r}\biggl(1+\|g(r)\|^{2})dr\nonumber\\ &\quad+C\lambda_{n+1}^{-\frac{1}{2}}\int_{\tau-t}^{\tau}e^{\gamma r}\|u(r,\tau-t,u_{\tau-t})\|_{H^{1}(\mathcal{O})}^{4}dr. \end{aligned} \tag{58}\]
By \(\|I-\mathcal{P}_n\|\leq1, \|\xi\|\leq1\) and \(B\in\mathcal{B}\), we obtain that as \(t\to+\infty,\) \[\begin{aligned} \label{eq3.41} e^{-\gamma t}\|(I-\mathcal{P}_n)(\xi u_{\tau-t})\|_{H^1}^2\leq Ce^{-\gamma t}\|u_{\tau-t}\|_{H^1(\mathcal{O})}^2\leq Ce^{-\gamma t}\|B\|_{H_0^1(\mathcal{O})}^2<\epsilon. \end{aligned} \tag{59}\]
Note that \(\lambda_{n+1}^{-\frac{1}{2}}\to0\) as \(n\to\infty.\) By [6], we have \[\begin{aligned} \label{eq3.42} \|v_{2}(\tau,\tau-t,v_{2,\tau-t})\|_{H^{1}(\mathcal{O}_{2k})}^{2}<\epsilon\quad\mathrm{as}\quad n,t\to+\infty. \end{aligned} \tag{60}\]
This completes the proof of Lemma 3.4. ◻
Lemma 4.1. Let \(\tau\in\mathbb{R},B\in\mathcal{B}_{\gamma},t_{m}\to\infty\) and \(u_{0m}\in B(\tau-t_{m}).\) If \(u_{m}(r,\tau-t_{m},u_{0m})\) is the solution of (1) with initial condition \(u_{0m}\) and
\[\begin{aligned} \label{eq4.1} v_m(r,\tau-t_m,u_{0m})=\xi\left(\frac{x_3^2}{k^2}\right)u_m(r,\tau-t_m,u_{0m}), \end{aligned} \tag{61}\] where \(k\geqslant1\). Then the sequence \(\{v_m(\tau,\tau-t_m,u_{0m})\}\) has a convergent subsequence in \(H_0^1(\mathcal{O})\).
Proof. Use Lemma 3.1, we get that
\[\begin{aligned} \label{eq4.2} \|u(\tau,\tau-t,u_0(\tau-t))\|_{H_0^1(\mathcal{O})}^2\leqslant\dfrac{2}{\nu\lambda}e^{-\gamma\tau}\int_{-\infty}^{\tau}e^{\gamma r}\|g(r)\|^2dr,\quad t\geqslant \widetilde{T}. \end{aligned} \tag{62}\]
By (62), we know
\[\begin{aligned} \label{eq4.3} \|u_m(\tau,\tau-t_m,u_{0m})\|_{H_0^1(\mathcal{O})}^2\leqslant\dfrac{2}{\nu\lambda}e^{-\gamma\tau}\int_{-\infty}^{\tau}e^{\gamma r}\|g(r)\|^2dr,\quad m\geqslant \widetilde{M}. \end{aligned} \tag{63}\]
From (61), we have
\[\begin{aligned} \label{eq4.4} \|v_m(\tau,\tau-t_m,u_{0m})\|_{H_0^1(\mathcal{O})}^2\leqslant ce^{-\gamma\tau}\int_{-\infty}^\tau e^{\gamma r}\|g(r)\|^2dr,\quad m\geqslant \widetilde{M}. \end{aligned} \tag{64}\]
Now, set \(\epsilon>0\) and \(k\geqslant1.\) By Lemma 3.4, there exist \(T_2=T_{2}(\tau,B,\epsilon)\) and \(N=N(\tau,\epsilon)\) such that \[\|(I-P_N)v(\tau,\tau-t,u_0(\tau-t))\|_{H_0^1(\mathcal{O}_{2k})}\leqslant\epsilon,\quad t\geqslant \widetilde{T}_{1},\]
so there is \(\widetilde{M}_1=\widetilde{M}_1(\tau,B,\epsilon)\) such that \(t_m\geqslant \widetilde{T}_{1}\) for \(m\geqslant \widetilde{M}_1.\)
\[\|(I-P_N)v_m(\tau,\tau-t_m,u_{0m})\|_{H_0^1(\mathcal{O}_{2k})}\leqslant\epsilon,\quad m\geqslant \widetilde{M}_1.\]
Since \(v_m(\tau,\tau-t_{m},u_{0m})=0\) on \(\mathcal{O}\setminus\mathcal{O}_{2k},\{v_{m}(\tau,\tau-\) \(t_m,u_{0m})\}\) is precompact in \(H_0^1(\mathcal{O}).\) ◻
Lemma 4.2. Let \(\tau\in\mathbb{R},B\in\mathcal{B}_{\gamma},t_{m}\to\infty\) and \(u_{0m}\in B(\tau-t_{m}),\) then \(u_{m}(\tau,\tau-t_{m},u_{0m})\) has a convergent subsequence in \(H_0^1(\mathcal{O})\).
Proof. See, for example [14]. ◻
It follow that the existence of a pullback attractor for the \(\theta\)-cocycle \(\Phi\).
Lemma 4.3. The problem (1) has a unique pullback \(\mathcal{B}_{\gamma}\)-attractor in \(H_{0}^{1}(\mathcal{O})\).
Proof. For \(\tau\in\mathbb{R}\), we get
\[(r_\gamma(\tau))^2=\dfrac{2}{\nu\lambda}\int_{-\infty}^{\tau}e^{\gamma r}\|g(r)\|^2dr,\] and \[D_\gamma(\tau)=\{u\in H_0^1(\mathcal{O}):\lVert u\rVert_{H_0^1(\mathcal{O})}\leqslant r_\gamma(\tau)\}.\]
Then, \(D_\gamma(\tau)\in\mathcal{B}_\gamma\) is pullback \(\mathcal{B}_\gamma\)-absorbing for \(\Phi\) in \(H_0^1(\mathcal{O})\) by Lemma 3.1. By Lemma 4.2, \(\Phi\) is pullback \(\mathcal{B}_\gamma\)-asymptotically compact in \(H_0^1.\) Thus, \(\Phi\) has a unique pullback \(\mathcal{B}_\delta\)-attractor from Theorem 2.7. ◻
The authors are grateful to the anonymous referees for their useful comments and suggestions. This work was supported by the Natural Science Foundation of Qinghai Province (No.2024-ZJ-931) and the Natural Science Foundation of China (No.12461039; 12161071).