Attractors for second-order evolution equations with a nonlinear damping
type: Item
instance of: scholarly article
title: Attractors for second-order evolution equations with a nonlinear damping
zbMATH Open document ID: 1072.37054
author: I. D. Chueshov
author: Irena Lasiecka
published in: Journal of Dynamics and Differential Equations
publication date: 2005-03-15T00:00:00Z
review text: Let \(\Omega\subset {\mathbb R}^n,n=2,3\), be a bounded connected set with a smooth boundary \(\Gamma\). The equation (1) \(\omega '-\Delta \omega +kg(\omega_t )+f(\omega )=0\) in \(Q=[0,\infty )\times \Omega\) with some boundary condition is considered. Here, \(g\in C^1 ({\mathbb R})\) is a monotone increasing function with some additional properties and \(f\in C^1 ({\mathbb R})\) is such that \(| f'(s)| \leq M| s| ^q, | s| \geq 1\) for some \(q\). It is shown that equation (1) generates a continuous semiflow \(S_t \) in \(H=H^1 (\Omega )\times L_2 (\Omega )\) and under some conditions, the dynamical system \((H,S_t )\) possesses a global compact attractor \({\mathcal A}\), under some hypotheses having a finite fractal dimension. There is a theorem dealing with the convergence to an equilibrium. Analogous results for a nonlinear system of dynamical elasticity described by von Karman evolution with a nonlinear dissipation are obtained. Main abstract results on global attractors are the following: 1. Let \((X,S_t )\) be a dissipative dynamical system in a Banach space \(X\). Then \((X,S_t )\) possesses a compact global attractor \({\mathcal A}\) iff \((X,S_t )\) is asymptotically smooth. 2. Criteria for asymptotic smoothness and criteria for finite dimensionality of attractors are obtained. 3. The long-time dynamics of abstract nonlinear second-order evolution equations with a nonlinear damping is studied. It is shown that any solution is stabilized to an equilibrium and the rate of convergence is estimated which in turn, depends on the behavior at the origin of the function describing the dissipation. If the damping is bounded from below by a linear function, this rate is exponential.
reviewed by: Victor Sharapov
Mathematics Subject Classification ID: 37L30
Mathematics Subject Classification ID: 34G20
Mathematics Subject Classification ID: 35B41
Mathematics Subject Classification ID: 35L70
zbMATH DE Number: 2145900
zbMATH Keywords: global attractors
zbMATH Keywords: dimension
zbMATH Keywords: nonlinear dissipation
zbMATH Keywords: smoothness
zbMATH Keywords: stabilization
zbMATH Keywords: equilibria
zbMATH Keywords: convergence rate
MaRDI profile type: Publication
full work available at URL: s10884-004-4289-x
OpenAlex ID: W2061391122
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DOI: 10.1007/S10884-004-4289-X
description: scientific article; zbMATH DE number 2145900