On the strongly damped wave equation
type: Item
instance of: scholarly article
title: On the strongly damped wave equation
zbMATH Open document ID: 1068.35077
author: Vittorino Pata
author: Marco Squassina
published in: Communications in Mathematical Physics
publication date: 2005-04-14T00:00:00Z
review text: The author considers the strongly damped wave equation \[ \begin{gathered} u_{tt}- \omega\Delta u_t-\Delta u+\phi(u)= f(t),\;x\in\Omega,\;t> 0,\;u(x,0)= u_0(x),\\ u_t(x,0)= u_1(x),\;u(x,t)= 0\quad\text{for }x\in \partial\Omega.\end{gathered}\tag{2} \] Here \(\Omega\subseteq \mathbb{R}^3\) is a smooth, bounded domain, and \(\omega> 0\). The emphasis of the paper is on the growth order of the nonlinearity which may be 5. A functional setting is imposed on (1), i.e. one sets \(A=-\Delta\) on \(\text{dom}(A)= H^2(\Omega)\cap H^1_0(\Omega)\) and then one defines the phase space of (1) in terms of the fractional power spaces \(D(A^s)\), i.e. one sets \(H_s= D(A^{(1+s)/2})\times D(A^{s/2})\). The assumptions on the nonlinearity are as follows: \[ |\phi(s)- \phi(r)|\leq c|r- s|(1+|r|^4+ |s|^4).\tag{1} \] One assumes that there is a decomposition \(\phi= \phi_0+\phi_1\), \(\phi_j\in C(\mathbb{R})\) such that \[ |\phi_0(r)|\leq c(1+|r|)^5,\;\phi_0(r)r\geq 0,\;|\phi_1(r)|\leq c(1+|r|^\gamma),\tag{3} \] some \(\gamma\in (0,5)\) and one also assumes \(\liminf r^{-1}\phi_1(r)> \alpha_1\) as \(|r|\to\infty\), where \(\alpha_1> 0\). Based on these assumptions, Theorem 1 asserts the existence of a unique global solution to (1), what entails the existence of a strongly continuous solution semigroup \(S(t)\), \(t> 0\), which has suitable Lipschitz properties by Theorem 2. Theorem 3 then asserts the existence of an absorbing set. Based on this fact, the existence of a universal attractor follows (Theorem 4). The author then considers a subcritical case which arises if \(f\in L^2(\Omega)\) is \(t\)-independent and where the \(\phi_j\) in (2) satisfy: \[ \phi_0= 0,\;\phi_1\in C^1(\mathbb R),\;|\phi_1'(r)|\leq c(1+|r|^{\gamma- 1}),\;r\in \mathbb R.\tag{4} \] Under assumptions (4), further properties of the global attractor can be deduced. In addition, the existence of exponential attractors can be proved, as is shown in the last part of the paper.
reviewed by: Bruno Scarpellini
Mathematics Subject Classification ID: 35L75
Mathematics Subject Classification ID: 37L30
Mathematics Subject Classification ID: 35B41
zbMATH DE Number: 2156004
zbMATH Keywords: growth order
zbMATH Keywords: functional setting
zbMATH Keywords: absorbing set
zbMATH Keywords: exponential attractors
MaRDI profile type: Publication
full work available at URL: s00220-004-1233-1
OpenAlex ID: W2150222152
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DOI: 10.1007/S00220-004-1233-1
description: scientific article; zbMATH DE number 2156004