On an extensible beam equation with nonlinear damping and source terms
type: Item
instance of: scholarly article
title: On an extensible beam equation with nonlinear damping and source terms
zbMATH Open document ID: 1329.35069
author: Zhijian Yang
published in: Journal of Differential Equations
publication date: 2013-04-25T00:00:00Z
review text: The paper is concerned with the global existence, stability and the longtime dynamics of solutions to the initial boundary value problem of an extensible beam equation with nonlinear damping and source terms, namely \[ u_{tt} - M(\|\nabla u\|^2)\Delta u + \Delta^2 u + g(u_t) + f (u) = h(x)\qquad \text{ in }\Omega\times \mathbb R^+, \] where \(\Omega\) is a bounded domain in \(\mathbb R^N\) with smooth boundary, either with hinged boundary condition \[ u|_{\partial\Omega}=0, \quad \Delta u|_{\partial\Omega}= 0, \] or clamped boundary condition \[ u|_{\partial\Omega} =0, \quad \frac{\partial u}{\partial \nu}\left|_{\partial\Omega}\right.= 0. \] It is proven that the problem is globally well posed, provided that either the growth exponent \(p\) of the source term \(f\) is non-supercritical, or that it is dominated by the growth exponent \(q\) of the nonlinear damping \(g\). Moreover, the related solution semigroup has a finite-dimensional global attractor and also an exponential attractor, provided either \(2\leq p\leq q\leq p^*\) or \(p = q > p^*\) if \(N\geq 5\). When the space dimension is less or equal than \(4\), then all the conclusions hold without any restrictions on \(p\) and \(q\).
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35B41
Mathematics Subject Classification ID: 74K10
Mathematics Subject Classification ID: 35L35
Mathematics Subject Classification ID: 35L76
Mathematics Subject Classification ID: 35R09
zbMATH DE Number: 6157572
zbMATH Keywords: global solutions
zbMATH Keywords: longtime dynamics
zbMATH Keywords: global attractor
zbMATH Keywords: exponential attractor
reviewed by: Pablo Amster
MaRDI profile type: Publication
full work available at URL: j.jde.2013.02.008
OpenAlex ID: W2051148538
DOI: 10.1016/J.JDE.2013.02.008
description: scientific article; zbMATH DE number 6157572