Exponential energy decay of solutions for an integro-differential equation with strong damping
type: Item
instance of: scholarly article
title: Exponential energy decay of solutions for an integro-differential equation with strong damping
zbMATH Open document ID: 1205.45012
author: Shun-Tang Wu
published in: Journal of Mathematical Analysis and Applications
publication date: 2010-02-19T00:00:00Z
review text: Consider \(\Omega\subset\mathbb R^N\), \(N\geq 1\), a bounded domain with a smooth boundary \(\partial\Omega\) so that the divergence theorem can be applied, the function \(M\in C^1([0,+\infty))\), \[ M(s)= m_0+ bs^\gamma \] with \(m_0\geq 0\), \(b\geq 0\), \(\gamma\geq 1\), a nondecreasing function \(g\in C^1([0,+\infty))\) so that \[ (\forall s\geq 0)(g(s)\geq 0),\quad m_0- \int^{+\infty}_0 g(s)\,ds= 1>0, \] a nonlinear function \(f(u)= |u|^{p-2}u\), \(p> 2\), where \(u: \Omega\times [0,+\infty)\to \mathbb R\). The author studies the initial boundary value problem for the integro-differential equation \[ \begin{gathered} u_{tt}- M(\|\nabla u\|^2_2)\Delta u+\int^t_0 g(t- s)\Delta u(s)\,ds-\Delta u_t= f(u),\\ u(x,0)= u_0(x),\quad u_t(x,0)= u_1(x),\quad x\in\Omega,\\ u(x,t)= 0,\quad x\in\partial\Omega,\quad t> 0.\end{gathered}\tag{1} \] Using the Sobolev space \(W^{m,p}(\Omega)\) and \(H^1_0(\Omega)= \overline{C^\infty_0(\Omega)}\) with respect to the norm \[ \| u\|_{H^1_0}=\|\nabla u\|_2, \] the author proves that if \(u_0\in H^1_0(\Omega)\cap W^{2,2}(\Omega)\), \(u_1\in L^2(\Omega)\), then there exists a unique solution \(u\) of the problem (1) satisfying \[ u\in C([0,T], H^1_0(\Omega)\cap W^{2,2}(\Omega)), \] \[ u_t\in C([0, T], L^2(\Omega)\cap L^2([0,T], H^1_0(\Omega))). \]
Mathematics Subject Classification ID: 45K05
Mathematics Subject Classification ID: 45M05
zbMATH DE Number: 5673245
zbMATH Keywords: asymptotic behavior
zbMATH Keywords: decay rate
zbMATH Keywords: strong damping
zbMATH Keywords: integro-differential equation
Wikidata QID: Q115346199
reviewed by: Dan-Mircea Borş
MaRDI profile type: Publication
full work available at URL: j.jmaa.2009.11.046
OpenAlex ID: W2005427652
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DOI: 10.1016/J.JMAA.2009.11.046
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description: scientific article; zbMATH DE number 5673245