Finite dimensionality and compactness of attractors for von Karman equations with nonlinear dissipation
type: Item
instance of: scholarly article
title: Finite dimensionality and compactness of attractors for von Karman equations with nonlinear dissipation
zbMATH Open document ID: 0960.35009
published in: NoDEA. Nonlinear Differential Equations and Applications
publication date: 2001-05-10T00:00:00Z
review text: The author proves the existence of a global, compact attractor of finite Hausdorff dimension for von Karmans shell equation with nonlinear damping: \[ w_{tt}+ \Delta^2w+ g(w_t)+ \rho\Delta w= [{\mathcal F}_f(w)+ \theta, w+f]+ F\tag{1} \] in \(Q= \Omega\times (0,\infty)\), with boundary conditions \(w= {\partial w\over\partial n}= 0\) on \(\partial\Omega\times (0,\infty)\), i.e. with \(w\in H^4(\Omega)\cap H^2_0(\Omega)\). Here \(\Delta^2{\mathcal F}_f(w)= -[w+ 2f,w]\) on \(Q\), \({\mathcal F}_f(w)\in H^4(\Omega)\cap H^2_0(\Omega)\), with \([ ,\;]\) the von Karman bracket: \([u,v]= u_{xx} v_{yy}+ u_{yy}v_{xx}- 2u_{xy} v_{xy}\). Finally, \(g: \mathbb{R}\to\mathbb{R}\) is a nonlinear damping, subject to \(0< m\leq g'(s)\), \(s\in\mathbb{R}\), and further conditions. The function \(f\in H^2_0(\Omega)\) is given and depends on the curvature of the shell, \(\theta\in H^2_0(\Omega)\cap W^{2,\infty}(\Omega)\) is the stress function due to thermal effects, while \(F\in L^2(\Omega)\) is an outer force. The proof of the existence of the global attractor turns out to be rather difficult, since many of the standard devices from attractor theory do not work in the present context, due to the nature of the nonlinearity. Theorem 1.1, proved in [\textit{A. Favini}, \textit{M. A. Horn}, \textit{I. Lasiecka} and \textit{D. Tataru}, Differ. Integral Equ. 9, No. 2, 267-294 (1996; Zbl 0847.35070)], serves as basis since it asserts global existence and uniqueness of (weak) solutions and thus provides the nonlinear semigroup \(T(t)\), \(t\geq 0\), describing the dynamics of (1). The main result is provided by Theorem 1.2, (I)--(III) which has to be summarized. Part (I) asserts the existence of an absorbing set. Part (II) states that if in addition also \[ g'(s)\leq Ms^2,\quad|s|\geq 1 \] holds, then a global compact attractor exists. Part (III) finally states that if \(g'(s)\) satisfies an additional smoothness condition then the Hausdorff dimension \(A\) is finite. The proof of Theorem 1.…
Mathematics Subject Classification ID: 35B41
Mathematics Subject Classification ID: 35L75
Mathematics Subject Classification ID: 35L55
Mathematics Subject Classification ID: 35L35
Mathematics Subject Classification ID: 74K25
Mathematics Subject Classification ID: 35Q72
Mathematics Subject Classification ID: 37L30
zbMATH DE Number: 1406553
zbMATH Keywords: nonlinear damping
zbMATH Keywords: global attractor
author: Irena Lasiecka
reviewed by: Bruno Scarpellini
MaRDI profile type: Publication
full work available at URL: s000300050012
OpenAlex ID: W1983821554
DOI: 10.1007/S000300050012
description: scientific article; zbMATH DE number 1406553