\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2014 (2014), No. 74, pp. 1--11.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu \newline ftp ejde.math.txstate.edu} \thanks{\copyright 2014 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2014/74\hfil Pullback attractor] {Pullback attractor for non-autonomous $p$-Laplacian equations with dynamic flux boundary conditions} \author[B. You, F. Li \hfil EJDE-2014/74\hfilneg] {Bo You, Fang Li} \address{Bo You \newline School of Mathematics and Statistics, Xi'an Jiaotong University\\ Xi'an, 710049, China} \email{youb03@126.com} \address{Fang Li \newline Department of Mathematics, Nanjing University\\ Nanjing, 210093, China} \email{lifang101216@126.com} \thanks{Submitted June 26, 2013. Published March 18, 2014.} \subjclass[2000]{35B40, 37B55} \keywords{Pullback attractor; Sobolev compactness embedding; $p$-Laplacian; \hfill\break\indent norm-to-weak continuous process; asymptotic a priori estimate; non-autonomous; \hfill\break\indent nonlinear flux boundary conditions} \begin{abstract} This article studies the long-time asymptotic behavior of solutions for the non-autonomous $p$-Laplacian equation \[ u_t-\Delta_pu+ |u|^{p-2}u+f(u)=g(x,t) \] with dynamic flux boundary conditions \[ u_t+|\nabla u|^{p-2}\frac{\partial u}{\partial\nu}+f(u)=0 \] in a $n$-dimensional bounded smooth domain $\Omega$ under some suitable assumptions. We prove the existence of a pullback attractor in $\big(W^{1,p}(\Omega)\cap L^q(\Omega)\big)\times L^q(\Gamma)$ by asymptotic a priori estimate. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \allowdisplaybreaks \section{Introduction} We are concerned with the existence of a pullback attractor in $\big(W^{1,p}(\Omega)\cap L^q(\Omega)\big)\times L^q(\Gamma)$ for the process $\{U(t,\tau)\}_{t\geq\tau}$ associated with solutions of the following non-autonomous $p$-Laplacian equation \begin{equation}\label{1} u_t-\Delta_pu+ |u|^{p-2}u+f(u)=g(x,t),\quad (x,t)\in\Omega\times \mathbb{R}_{\tau}. \end{equation} This equation is subject to the dynamic flux boundary condition \begin{equation}\label{2} u_t+|\nabla u|^{p-2}\frac{\partial u}{\partial\nu}+f(u)=0,\quad \quad (x,t)\in\Gamma\times\mathbb{R}_{\tau} \end{equation} and the initial conditions \begin{gather} \label{3} u(x,\tau)=u_\tau(x),\quad x\in\Omega,\\ \label{4} u(x,\tau)=\theta_\tau(x),\quad x\in\Gamma, \end{gather} where $\Omega \subset \mathbb{R}^n$ $(n\geq 3)$ is a bounded domain with smooth boundary $\Gamma$, $\nu$ denotes the outer unit normal on $\Gamma$, $p\geq 2$, $\mathbb{R}_\tau=[\tau,+\infty)$, the nonlinearity $f$ and the external force $g$ satisfy some conditions, specified later. To study problem \eqref{1}-\eqref{4}, we assume the following conditions: \begin{itemize} \item[(H1)] the function $f\in C^1(\mathbb{R},\mathbb{R})$ and satisfies \begin{equation}\label{5} f'(u)\geq -l \end{equation} for some $l\geq 0$, and \begin{equation}\label{6} c_1|u|^q-k\leq f(u)u\leq c_2|u|^q+k, \end{equation} where $c_i> 0$ ($i=1,2$), $q\geq 2$, $ k> 0$. \item[(H2)] The external force $g:\Omega\times \mathbb{R}\to \mathbb{R}$ is locally Lipschitz continuous, $ g$ belongs to $H_{\rm loc}^1(\mathbb{R}, L^2(\Omega))$, and satisfies \begin{equation}\label{7} \int_{-\infty}^te^{c_1 s}\|g(s)\|_{L^2(\Omega)}^2\,ds +\int_{-\infty}^te^{c_1 s}\|g_t(s)\|_{L^2(\Omega)}^2\,ds<\infty \end{equation} for all $t\in\mathbb{R}$. \end{itemize} Dynamic boundary conditions are very natural in many mathematical models such as heat transfer in a solid in contact with a moving fluid, thermoelasticity, diffusion phenomena, heat transfer in two medium, problems in fluid dynamics (see \cite{ ajm, ajm1, bar, ca, ca1, fzh, pj, pl, yl, yl1}). The understanding of the asymptotic behavior of dynamical systems is one of the most important problems of modern mathematical physics. One way to treat this problem for a dissipative system is to analyze the existence and structure of its attractor. Generally speaking, the attractor has a very complicated geometry which reflects the complexity of the long-time behavior of the system. There are many authors who have considered the long-time behavior of solutions for the problems of dynamic boundary conditions. For example, the authors considered the existence of global attractors, respectively, in $L^2(\bar{\Omega},d\mu)$, $L^q(\bar{\Omega},d\mu)$ and $\big(H^1(\Omega)\cap L^q(\Omega)\big)\times L^q(\Gamma)$ for the reaction-diffusion equation with dynamic flux boundary conditions in \cite{fzh}. The existence of uniform attractors in $L^2(\bar{\Omega},d\mu)$ and $\big(H^1(\Omega)\cap L^q(\Omega)\big)\times L^q(\Gamma)$ for the reaction-diffusion equation with dynamic flux boundary conditions was proved in \cite{yl}. In \cite{yb}, the authors proved the existence of global attractors for the autonomous $p$-Laplacian equation with dynamic flux boundary conditions in $L^2(\bar{\Omega},d\mu)$, $L^q(\bar{\Omega},d\mu)$ by the Sobolev compactness embedding theorem and the existence of a global attractor for the autonomous $p$-Laplacian equation with dynamic flux boundary conditions in $\big(W^{1,p}(\Omega)\cap L^q(\Omega)\big)\times L^q(\Gamma)$ by asymptotical a priori estimate. Recently, the existence of uniform attractors in $L^2(\bar{\Omega},d\mu)$ and $\big(W^{1,p}(\Omega)\cap L^q(\Omega)\big)\times L^q(\Gamma)$ for the non-autonomous $p$-Laplacian equation with dynamic flux boundary conditions was obtained in \cite{lk}. Non-autonomous equations appear in many applications in natural sciences, so they are of great importance and interest. The long-time behavior of solutions for the non-autonomous equations has been studied extensively in recent years (see \cite{ dn,hc, hc1,ct, pe, pe1, lss, bs, yl}). For instance, the existence of a pullback attractor in $L^2(\Omega)$ was studied in \cite{ct1}. The authors obtained the existence of a pullback attractor in $H_0^1(\Omega)$ in \cite{sht1}. The existence of a pullback attractor in $H_0^1(\Omega)$ was considered in \cite{ly}. The authors proved the existence of a pullback attractor in $L^p(\Omega)$ for a reaction-diffusion equation in \cite{ly1} under the assumption \begin{equation*} \|g(s)\|_2^2\leq Me^{\alpha |s|} \end{equation*} for all $s\in\mathbb{R}$ and $0\leq\alpha<\lambda_1$, where $\lambda_1$ is the first eigenvalue of $-\Delta$ with Dirichlet boundary condition. In \cite{yl1}, the authors used a new type of uniform Gronwall inequality and proved the existence of a pullback attractor in $L^{r_1}(\Omega)\times L^{r_2}(\Gamma)$ for the equation \begin{gather*} u_t-\Delta_pu+ |u|^{p-2}u+f(u)=h(t),\quad (x,t)\in\Omega\times\mathbb{R}_\tau,\\ u_t+|\nabla u|^{p-2}\frac{\partial u}{\partial\nu}+g(u)=0,\quad (x,t)\in\Gamma\times\mathbb{R}_\tau,\\ u(x,\tau)=u_0(x),\quad x\in\bar{\Omega} \end{gather*} under the assumptions that $f$, $g$ satisfy the polynomial growth condition with orders $r_1$, $r_2$ and $\|h(t)\|_{L^2(\Omega)}$ satisfies some weak assumption \begin{equation*} \int_{-\infty}^te^{\theta s}\|h(s)\|_{L^2(\Omega)}^2\,ds<\infty \end{equation*} for all $t\in\mathbb{R}$, where $\theta$ is some positive constant. By using their main result, we can get the following result. \begin{corollary}\label{51} Let $\Omega$ be a bounded domain in $\mathbb{R}^n$ with smooth boundary $\Gamma$, let $f$ and $g$ satisfy {\rm (H1)--(H2)}. Then the process $\{U(t,\tau)\}_{t\geq \tau}$ corresponding to \eqref{1}-\eqref{4} has a pullback $\mathcal{D}$-attractor $\mathcal{A}_q$ in $L^q(\bar{\Omega},d\mu)$, which is pullback $\mathcal{D}$-attracting in the topology of $L^q(\bar{\Omega},d\mu)$-norm. \end{corollary} The study of non-autonomous dynamical systems is an important subject, it is necessary to study the existence of pullback attractors for the non-autonomous $p$-Laplacian equation with dynamic flux boundary conditions. Nevertheless, there are few results about the existence of a pullback attractor in $(W^{1,p}(\Omega)\cap L^q(\Omega))\times L^q(\Gamma)$ for the non-autonomous $p$-Laplacian equation with dynamic flux boundary conditions. The main difficulty is that in our case of the equation with $p$-Laplacian operator for $p>2$, we cannot use $-\Delta u_2$ as the test function to verify pullback $\mathcal{D}$-condition, which increases the difficulty in getting an appropriate form of compactness. To overcome this difficulty, we combine the idea of norm-to-weak process with asymptotic a priori estimates to prove the existence of a pullback attractor for the non-autonomous $p$-Laplacian equation with dynamic flux boundary conditions in $\big(W^{1,p}(\Omega)\cap L^q(\Omega)\big)\times L^q(\Gamma)$. The main purpose of this paper is to study the existence of a pullback attractor in $(W^{1,p}(\Omega)\cap L^q(\Omega))\times L^q(\Gamma)$ for the non-autonomous $p$-Laplacian evolutionary equation \eqref{1}-\eqref{4} under quite general assumptions \eqref{5}-\eqref{7}. Here, we state our main result as follows. \begin{theorem}\label{main} Assume that {\rm (H1)--(H2)} hold. Then the process $\{U(t,\tau)\}_{t\geq \tau}$ corresponding to problem \eqref{1}-\eqref{4} has a pullback $\mathcal{D}$-attractor $\mathcal {A}$ in $(W^{1,p}(\Omega)\cap L^q(\Omega))\times L^q(\Gamma)$. \end{theorem} This article is organized as follows: In the next section, we give some notation and lemmas used in the sequel. Section 3 is devoted to proving the existence of a pullback absorbing set in $\big(L^2(\Omega)\cap W^{1,p}(\Omega)\cap L^q(\Omega)\big) \times\big(L^2(\Gamma)\cap L^q(\Gamma)\big)$ and the existence of a pullback attractor in $\big(L^q(\Omega)\cap W^{1,p}(\Omega)\big)\times L^q(\Gamma)$. Throughout this paper, let $C$ be a positive constant, which may be different from line to line (and even in the same line), we denote the trace of $u$ by $v$. \section{Preliminaries} To study \eqref{1}-\eqref{4}, we recall the Sobolev space $W^{1,p}(\Omega)$ defined as the closure of $C^{\infty}(\Omega)\cap W^{1,p}(\Omega)$ in the norm \begin{equation*} \|u\|_{1,p}=\Big(\int_{\Omega}|\nabla u|^p+|u|^p\,dx\Big)^{1/p} \end{equation*} and denote by $X^*$ the dual space of $X$. We also define the Lebesgue spaces as follows \begin{equation*} L^r(\Gamma)=\{v:\|v\|_{L^r(\Gamma)}<\infty\}, \end{equation*} where \begin{equation*} \|v\|_{L^r(\Gamma)}=\Big(\int_{\Gamma}|v|^r\,dS\Big)^{1/r} \end{equation*} for $r\in [1,\infty)$. Moreover, we have \begin{gather*} L^s(\Omega)\oplus L^s(\Gamma)=L^s(\bar{\Omega},d\mu),\quad s\in [1,\infty),\\ \|U\|_{L^s(\bar{\Omega},d\mu)}=\Big(\int_{\Omega}|u|^s\,dx\Big)^{1/s} +\Big(\int_{\Gamma}|v|^s\,dS\Big)^{1/s} \end{gather*} for any $U=\begin{pmatrix}u\\ v\end{pmatrix}\in L^s(\bar{\Omega},d\mu)$, where the measure $d\mu=dx|_{\Omega}\oplus dS|_{\Gamma}$ on $\bar{\Omega}$ is defined for any measurable set $A\subset \bar{\Omega}$ by $\mu(A)=|A\cap\Omega|+S(A\cap \Gamma)$. In general, any vector $\theta\in L^s(\bar{\Omega},d\mu)$ will be of the form $\begin{pmatrix} \theta_1 \\ \theta_2\end{pmatrix}$ with $\theta_1\in L^s(\Omega,dx)$ and $\theta_2\in L^s(\Gamma,dS)$, and there need not be any connection between $\theta_1$ and $\theta_2$. \begin{remark}[\cite{gcg}] \label{80} \rm $C(\bar{\Omega})$ is a dense subspace of $L^2(\bar{\Omega},d\mu)$ and a closed subspace of $L^{\infty}(\bar{\Omega},d\mu)$. \end{remark} Next, we recall briefly some lemmas used to prove the existence of pullback absorbing sets for \eqref{1}-\eqref{4} under some suitable assumptions. \begin{lemma}[\cite{bt}] \label{11} Let $x,y\in\mathbb{R}^n$ and let $\langle\cdot,\cdot\rangle$ be the standard scalar product in $\mathbb{R}^n$. Then for any $p\geq 2$, there exist two positive constants $C_1$, $C_2$ which depend on $p$ such that \begin{gather*} \langle|x|^{p-2}x-|y|^{p-2}y,x-y\rangle\geq C_1|x-y|^p,\\ \big|\,|x|^{p-2}x-|y|^{p-2}y\big|\leq C_2(|x|+|y|)^{p-2}|x-y|. \end{gather*} \end{lemma} \section{Existence of pullback attractors} In this section, we prove the existence of pullback attractors of solutions for problem \eqref{1}-\eqref{4}. \subsection{Well-posedness of solutions for problem \eqref{1}-\eqref{4}} In this subsection, we give the well-posedness of solutions for problem \eqref{1}-\eqref{4} which can be obtained by the Faedo-Galerkin method (see \cite{tr}). Here, we only state it as follows. \begin{theorem}\label{14} Under the assumptions {\rm (H1)--(H2)}, for any initial data $(u_\tau,\theta_\tau)\in L^2(\bar{\Omega},d\mu)$, there exists a unique weak solution $u(x,t)\in C(\mathbb{R}_\tau;L^2(\bar{\Omega},d\mu))$ of problem \eqref{1}-\eqref{4} and the mapping \begin{equation*} (u_\tau,\theta_\tau)\to (u(t),v(t)) \end{equation*} is continuous on $L^2(\bar{\Omega},d\mu)$. \end{theorem} By Theorem \ref{14}, we can define a family of continuous processes $\{U(t,\tau):-\infty<\tau\leq t<\infty\}$ in $L^2(\bar{\Omega},d\mu)$ as follows: for all $t\geq\tau$, \begin{equation*} U(t,\tau)(u_\tau,\theta_\tau)=(u(t),v(t)) :=(u(t;\tau,(u_\tau,\theta_\tau)),v(t;\tau,(u_\tau,\theta_\tau))), \end{equation*} where $u(t)$ is the solution of problem \eqref{1}-\eqref{4} with initial data $(u(\tau),v(\tau))=(u_\tau, \theta_\tau) \in L^2(\bar{\Omega},d\mu)$. That is, a family of mappings $U(t,\tau): L^2(\bar{\Omega},d\mu)\to L^2(\bar{\Omega},d\mu)$ satisfies \begin{gather*} U(\tau,\tau)=id\quad \text{(identity)},\\ U(t,\tau)=U(t,r)U(r,\tau)\quad\text{for all }\tau\leq r\leq t. \end{gather*} \subsection{Existence of a pullback absorbing set} In this subsection, we recall some basic definitions and abstract results about pullback attractors. \begin{definition}[\cite{ly, yl}] \rm Let $X$ be a Banach space. A process $\{U(t,\tau)\}_{t\geq\tau}$ is said to be norm-to-weak continuous on $X$, if for any $t$, $\tau\in\mathbb{R}$ with $t\geq\tau$ and for every sequence $x_n\in X$, from the condition $x_n\to x$ strongly in $X$, it follows that $U(t,\tau)x_n\to U(t,\tau)x$ weakly in $X$. \end{definition} \begin{lemma}[\cite{ly, yl}] \label{33} Let $X$ and $Y$ be two Banach spaces, and let $X^{*}$ and $Y^{*}$ be the dual spaces of $X$ and $Y$, respectively. If $X$ is dense in $Y$, the injection $i: X\to Y$ is continuous and its adjoint $i^{*}: Y^{*}\to X^{*}$ is dense. In addition, assume that $\{U(t,\tau)\}_{t\geq \tau}$ is a continuous or weak continuous process on $Y$. Then $\{U(t,\tau)\}_{t\geq \tau}$ is a norm-to-weak continuous process on $X$ if and only if $\{U(t,\tau)\}_{t\geq \tau}$ maps compact sets of $X$ into bounded sets of $X$ for any $t$, $\tau\in\mathbb{R}$, $t\geq\tau$. \end{lemma} Let $\mathcal{D}$ be a nonempty class of families $\hat{D}=\{D(t):t\in\mathbb{R}\}$ of nonempty subsets of $X$. \begin{definition}[\cite{ct}] \rm The process $\{U(t,\tau)\}_{t\geq\tau}$ is said to be pullback $\mathcal{D}$-asymp\-totically compact, if for any $t\in\mathbb{R}$ and any $\hat{D}\in\mathcal{D}$, any sequence $\tau_n\to -\infty$ and any sequence $x_n\in D(\tau_n)$, the sequence $\{U(t,\tau_n)x_n\}_{n=1}^{\infty}$ is relatively compact in $X$. \end{definition} \begin{definition}[\cite{yl}] \rm A family $\hat{\mathcal{A}}=\{A(t):t\in\mathbb{R}\}$ of nonempty subsets of $X$ is said to be a pullback $\mathcal{D}$-attractor for the process $\{U(t,\tau)\}_{t\geq\tau}$ in $X$, if \begin{itemize} \item [(i)] $A(t)$ is compact in $X$ for any $t\in\mathbb{R}$, \item [(ii)] $\hat{\mathcal{A}}$ is invariant, i.e., $U(t,\tau)A(\tau)=A(t)$ for any $\tau\leq t$, \item [(iii)] $\hat{\mathcal{A}}$ is pullback $\mathcal{D}$-attracting, i.e., \begin{equation*} \lim_{\tau\to -\infty}\operatorname{dist}(U(t,\tau)D(\tau),A(t))=0 \end{equation*} for any $t\in\mathbb{R}$ and any $\hat{D}=\{D(t):t\in\mathbb{R}\}\in\mathcal{D}$. \end{itemize} Such a family $\hat{\mathcal{A}}$ is called minimal if $A(t)\subset C(t)$ for any family $\hat{C}=\{C(t):t\in\mathbb{R}\}$ of closed subsets of $X$ such that $\lim_{\tau\to -\infty}\operatorname{dist}(U(t,\tau)B(\tau),C(t))=0$ for any $\hat{B}=\{B(t):t\in\mathbb{R}\}\in\mathcal{D}$. \end{definition} \begin{definition}[\cite{ct, yl}] \rm It is said that $\hat{B}\in\mathcal{D}$ is pullback $\mathcal{D}$-absorbing for the process $\{U(t,\tau)\}_{t\geq\tau}$, if for any $\hat{D}\in\mathcal{D}$ and any $t\in\mathbb{R}$, there exists a $\tau_0(t,\hat{D})\leq t$ such that $U(t,\tau)D(\tau)\subset B(t)$ for any $\tau\leq\tau_0(t,\hat{D})$. \end{definition} \begin{lemma}[\cite{ct, ly, yl}] \label{34} Let $\{U(t,\tau)\}_{t\geq\tau}$ be a process in $X$ satisfying the following conditions: \begin{itemize} \item [(1)] $\{U(t,\tau)\}_{t\geq\tau}$ be norm-to-weak continuous in $X$. \item [(2)] There exists a family $\hat{B}$ of pullback $\mathcal{D}$-absorbing sets $\{B(t):t\in\mathbb{R}\}$ in $X$. \item [(3)] $\{U(t,\tau)\}_{t\geq\tau}$ is pullback $\mathcal{D}$-asymptotically compact. \end{itemize} Then there exists a minimal pullback $\mathcal{D}$-attractor $\hat{\mathcal{A}}=\{A(t):t\in\mathbb{R}\}$ in $X$ given by \begin{equation*} A(t)= \cap_{s\leq t}\overline{\cup_{\tau\leq s}U(t,\tau)B(\tau)}. \end{equation*} \end{lemma} \begin{lemma}[\cite{yl}]\label{36} Suppose that \begin{equation*} y'(s)+\delta y(s)\leq b(s) \end{equation*} for some $\delta>0$, $t_0\in\mathbb{R}$ and for any $s\geq t_0$, where the functions $y$, $y'$, $b$ are assumed to be locally integrable and $y$, $b$ are nonnegative on the interval $t