Attractors for strongly damped wave equations with critical nonlinearities.
type: Item
instance of: scholarly article
title: Attractors for strongly damped wave equations with critical nonlinearities.
zbMATH Open document ID: 1060.35082
published in: Pacific Journal of Mathematics
publication date: 2004-09-16T00:00:00Z
review text: The authors consider the following strongly damped wave equation in a bounded smooth domain \(\Omega\) of \(\mathbb R^n\): \[ \begin{aligned} u_{tt}+\eta(-\Delta)^\theta u_t-\Delta u=f(u),&\quad x\in\Omega,\\ u(0,x)=u_0(x),\;\;u_t(0,x)=v_0(x),&\quad x\in\Omega,\tag{1}\\ u(t,x)=0, &\quad x\in\partial\Omega, \end{aligned} \] under the assumptions that \(\theta\in[1/2,1]\) and \(f\) is dissipative and satisfies the following growth restriction: \[ | f(u_1)-f(u_2)| \leq c| u_1-u_2| (1+| u_1| ^{\rho-1}+| u_2| ^{\rho-1}),\;\;u_1,u_2\in\mathbb R \tag{2} \] with \(\rho\leq\frac{n+2}{n-2}\). The main result of the paper is the existence of a compact global attractor (in the standard energy phase space) in the following cases: 1) \(\theta\in[1/2,1)\), \(\rho<\frac{n+2}{n-2}\). 2) \(\theta=1\), \(\rho\leq\frac{n+2}{n-2}\). The proof is strongly based on the fact that the linear part of (1) generates an analytic semigroup and on the previous results of the authors on the local well-posedness of (1) [see the authors, Bull. Aust. Math. Soc. 66, 443--463 (2002; Zbl 1020.35059)].
reviewed by: S. V. Zelik
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 37L30
Mathematics Subject Classification ID: 35L20
Mathematics Subject Classification ID: 35B41
Mathematics Subject Classification ID: 35L82
zbMATH DE Number: 2101023
zbMATH Keywords: strongly damped wave equation
zbMATH Keywords: critical nonlinearity
zbMATH Keywords: global attractor
zbMATH Keywords: compact global attractor
zbMATH Keywords: analytic semigroup
author: A. N. Carvalho
author: Jan W. Cholewa
MaRDI profile type: Publication
full work available at URL: pjm.2002.207.287
OpenAlex ID: W2071595923
DOI: 10.2140/PJM.2002.207.287
Recommended article: Attractors for strongly damped wave equations
Recommended article: A non-autonomous strongly damped wave equation: existence and continuity of the pullback attractor
Recommended article: Strongly damped wave equation in uniform spaces
Recommended article: Q4235765
Recommended article: Global attractors for strongly damped wave equations with subcritical-critical nonlinearities
Recommended article: Strongly damped wave equation with exponential nonlinearities
Recommended article: Existence and continuity of the attractor for a singularly perturbed hyperbolic equation
Recommended article: Q3519737
Recommended article: Q3381255
Recommended article: Pullback \(\mathcal D\)-attractors for nonautonomous sine-Gordon equations
description: scientific article; zbMATH DE number 2101023