Random attractor for a damped sine-Gordon equation with white noise
type: Item
instance of: scholarly article
title: Random attractor for a damped sine-Gordon equation with white noise
zbMATH Open document ID: 1065.37057
author: Xianqiang Yang
published in: Pacific Journal of Mathematics
publication date: 2005-02-24T00:00:00Z
review text: It is shown that a sine-Gordon equation with additive white noise, formally given by \[ u_{tt}+\alpha u_t-\Delta u+\beta\sin u=q\dot W \] on an open bounded \(\Omega\subset\mathbb R^n\) with smooth boundary, where \(\alpha>0\), \(q\in H^2(\Omega)\cap H_0^1(\Omega)\), \(\dot W\) is the formal derivative of a one-dimensional Wiener process, imposing Dirichlet conditions, has a random attractor. A nonrandom upper bound for the Hausdorff dimension of the random attractor, which decreases as the damping \(\alpha\) grows, is derived. The Hausdorff dimension of random attractors has been shown to be nonrandom almost surely by \textit{H. Crauel} and \textit{F. Flandoli} [J. Dyn. Differ. Equations 10, 449-474 (1998; Zbl 0927.37031)].
reviewed by: Hans Crauel
Mathematics Subject Classification ID: 37L30
Mathematics Subject Classification ID: 35R60
Mathematics Subject Classification ID: 35B41
Mathematics Subject Classification ID: 35Q53
Mathematics Subject Classification ID: 37L55
Mathematics Subject Classification ID: 60H15
zbMATH DE Number: 2138435
zbMATH Keywords: random attractor
zbMATH Keywords: sine-Gordon equation
zbMATH Keywords: Hausdorff dimension
MaRDI profile type: Publication
full work available at URL: pjm.2004.216.63
OpenAlex ID: W2015341875
DOI: 10.2140/PJM.2004.216.63
description: scientific article; zbMATH DE number 2138435