Pullback attractors for a semilinear heat equation on time-varying domains
type: Item
instance of: scholarly article
title: Pullback attractors for a semilinear heat equation on time-varying domains
zbMATH Open document ID: 1173.37064
author: Peter E. Kloeden
author: Chunyou Sun
author: José Real
published in: Journal of Differential Equations
publication date: 2009-06-10T00:00:00Z
review text: The authors study a semilinear heat equation on a domain which varies with time. To be more explicit, let \({\mathcal O}\subseteq \mathbb R^N\) be nonempty bounded, with \(C^2\)-boundary \(\partial{\mathcal O}\), let \(r\in C^1(\overline{\mathcal O}\times \mathbb R : \mathbb R^N)\) be a vector function such that \[ r(\cdot,t):{\mathcal O}\to {\mathcal O}_t\tag{1} \] is a \(C^2\)-diffeomorphism for all \(t\in \mathbb R\). For \(\tau<T\) set \[ \begin{alignedat}{2}2 Q_{\tau T} &=\bigcup_{t\in(\tau,T)}{\mathcal O}_t\times \{t\}, &\quad Q_\tau&= Q_{\tau\infty}\\ \Sigma_{\tau T} &=\bigcup_{t\in(\tau,T)}\partial{\mathcal O}_t\times \{t\}, &\quad \Sigma_\tau&=\Sigma_{\tau\infty}.\end{alignedat} \] We also set \(F(\cdot,t)=r^{-1}(\cdot,t)\). The equation in question is \[ \partial u_t-\Delta u+g(u)=f(t)\text{ in }Q_\tau, \quad u=0\text{ on }\Sigma_\tau, \quad\text{and}\quad u(\tau,x)=u_\tau(x),\qquad x\in{\mathcal O}_\tau.\tag{2} \] \((2)_T\) is the problem obtained from (2) by replacing \(Q_\tau\) and \(\Sigma_\tau\) by \(Q_{\tau T}\) and \(\Sigma_{\tau T}\), respectively. Here \(u_\tau:{\mathcal O}_\tau\to\mathbb R\), \(f:Q_\tau\to \mathbb R\) and \(g\in C^1(R,R)\), such that \[ -\beta+\alpha_1|s|^p\leq sg(s)\leq \beta+\alpha_2|s|^p, \] some \(\beta,\alpha_1,\alpha_2\geq 0\), \(p\geq 2\), and \(g'(s)\geq -\ell\), for \(s\in R\) and some \(\ell\geq 0\). In order to interprete (2), \((2)_T\) in a suitable functional setting, a considerable amount of preparatory steps are needed. Among others, strong and weak solutions are defined. The first major result is provided by Theorem 4.4 which asserts the following. Let \(f\in{\mathcal L}^2_{\text{loc}}(\mathbb R;{\mathcal L}^2({\mathcal O}_t))\). Let either \(\partial{\mathcal O}\) be \(C^2\) and \(N\leq 2p/p-2\) or let else \(\partial{\mathcal O}\) be \(C^m\) with \(m\geq N(p-2)/2p\). Then, for any \(u_\tau\in H^1_0({\mathcal O}_\tau)\cap{\mathcal L}^p({\mathcal O}_\tau)\) and \(\tau<T<\infty\) there exists a unique strong soluti…
reviewed by: Bruno Scarpellini
Mathematics Subject Classification ID: 37L30
Mathematics Subject Classification ID: 35Q30
Mathematics Subject Classification ID: 35K90
zbMATH DE Number: 5563803
zbMATH Keywords: semilinear heat equation
zbMATH Keywords: time-varying domain
zbMATH Keywords: nonautonomous dynamical system
zbMATH Keywords: pullback attractor
MaRDI profile type: Publication
full work available at URL: j.jde.2008.11.017
OpenAlex ID: W2018191090
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DOI: 10.1016/J.JDE.2008.11.017
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description: scientific article; zbMATH DE number 5563803