Asymptotically autonomous dynamics for parabolic equations
Let \(X\) be a Banach space, \(S: (t,s)\to S(t,s)\in X,\) for \(t\geq s\), be an evolution process with a pullback attractor \(\mathcal A=\{\mathcal A(t):t\in \mathbb R\} \) and \(T\) a semigroup with a global attractor \(\mathcal A_{\infty}\) on \(X\). The process is assumed to be continuous, that is the mapping \((t,x)\to S(t,s)x\) is continuous from \([s,+\infty)\times X\to X\) for each \(s\in\mathbb R\). The following results are obtained. Suppose \(S\) is (weakly) asymptotically autonomous to \(T\). Then the semi-continuity \[ \mathop{\text{dist}_X}_{\tau\to +\infty} (\mathcal A(\tau),\mathcal A_{\infty})=0 \] holds true if and only if \(\mathcal A\) is forward compact, i.e., \(\bigcup_{s\geq\tau}\mathcal A (s)\) is precompact for each \(\tau\in\mathbb R\) (Theorems 1.1 and 2.4). Some applications are given: {\parindent=6mm\begin{itemize}\item[1.] for the abstract equation \[ u_t+A(t)u=B(t),\quad t\geq \tau,\quad u(\tau)=u_{\tau} \] in \(L^2(\Omega)\), where \(\Omega\) is a bounded domain, and \[ A(t)u=-\text{div}\big(D(t,x)|\nabla u|^{p(x)-2}\nabla u\big)+|u|^{p(x)-2}u \] \item[2.] for the \(p\)-Laplace equation \[ u_t+Au+\lambda u+f(x,u)=g(t,x),\quad t\geq \tau,\quad x\in\mathbb R^n, \] \[ u(\tau)=u_{\tau}, \quad \tau\in\mathbb R, \] where \(\lambda>0 \) and \(A:W^{1,p}(\mathbb R^n)\to W^{-1,p^{\prime}}(\mathbb R^n)\) is the \(p\)-Laplace operator: \[ Au=-\sum_{i=1}^n \frac{\partial}{\partial x_i}(|u_{x_i}|^{p-2}u_{x_i}) , \] where \(p>2\). \end{itemize}} In each case, under suitable assumptions on the data, it is shown that the associated pullback attractor \(\mathcal A(t) \) converges, in some sense, as \(t\to\infty\), to the global attractor \(\mathcal A_{\infty}\), and that \(\mathcal A_{\infty}\supset \mathcal A(\infty),\) where \(\mathcal A(\infty)\) is the forward limit-set of the pullback attractor (Theorems 3.2. and 5.3.). The authors compare their results with recent results by \textit{P. E. Kloeden} and \textit{J. Simsen} [J. Math. Anal. Appl. 425, No. 2, 911--918 (2015; Zbl 1310.35203)].
scientific article; zbMATH DE number 6817619
scientific article