Global existence and asymptotic stability for viscoelastic problems
type: Item
instance of: scholarly article
title: Global existence and asymptotic stability for viscoelastic problems
zbMATH Open document ID: 1015.35071
published in: Differential and Integral Equations
publication date: 2003-07-20T00:00:00Z
review text: The authors investigate existence, uniqueness and decay rates of a solution of the semilinear viscoelastic wave equation \[ u_{tt}-\Delta u+\alpha u + f(u) + \int_0^t g(t-\tau)\Delta u(\tau) d\tau + h(u_t)=0 \quad \text{in} \Omega\times (0,\infty), \] \(u(x,0)=u_0(x)\), \(u_t(x,0)=u_1(x)\), \(x\in \Omega\), where \(\Omega\) is any bounded or finite measure domain of \(\mathbb R^n\). If \(\Omega\) has the nonempty boundary \(\Gamma\), then it is assumed to be regular only in the case of the boundary condition \(u=0\) on \(\Gamma\). The functions \(f, h\) are of the form \(f(s)=\gamma|s|^\xi s\), \(h(s)=\beta |s|^\rho s\), where \(0<\xi\), \(\rho\leq \frac 2{n-2}\) if \(n\geq 3\) and \(\Omega\) is bounded or \(\Omega\) is unbounded but possesses finite measure, \( 0<\xi=\rho= \frac 2{n-2}\) if \(n\geq 3\) and \(\Omega\) is unbounded, \(\xi,\rho>0\) if \(n=1,2\) and \(\Omega\) is arbitrary. The kernel function \(g\in C^2(\mathbb R_+)\) is bounded and satisfying \(1-\int_0^\infty g(s)ds= l>0\), \(-\xi_1g(t)\leq g'(t)\leq -\xi_2 g(t)\), \(0\leq g''(t)\leq \xi_3 g(t)\) \(\forall t\geq 0.\) Let \(\mathcal H\) is a Hilbert space endowed with the inner product \((u,v)_\mathcal H = (u,v)_{H^1_0(\Omega)}+(\Delta u,\Delta v).\) The main results are the following theorems: Theorem 2.1. Let \(\{u_0,u_1\}\in \mathcal H\times H^1_0(\Omega)\). Then the above problem possesses a unique regular solution in the class \(u,u'\in L^\infty(0,\infty;H^1_0(\Omega))\), \(u''\in L^\infty(0,\infty;L^2(\Omega))\). Moreover, if \(\|g\|_{L^1(0,\infty)}\) is sufficiently small and \(\Omega\) has finite measure, the energy \(E(t)=\frac 12\int_\Omega\{|u_t(x,t)|^2+|\nabla u(x,t)|^2+\alpha |u(x,t)|^2+\frac \gamma{\xi+2}|u(x,t)|^2\} dx\) has the decay rates \(E(t)\leq (\varepsilon\theta t+[E(0)]^{-\rho/2})^{-2/\rho}\), \(\forall t\geq 0\), \(\forall \varepsilon\in (0,\varepsilon_0]\), \(\theta>0\), \(\varepsilon_0>0.\) Theorem 2.2. Let \(\{u_0,u_1\}\in H^1_0(\Omega)\times L^2(\Omega)\). Then the above …
reviewed by: Igor Bock
Mathematics Subject Classification ID: 35L70
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 74D10
Mathematics Subject Classification ID: 93D20
zbMATH DE Number: 1839927
zbMATH Keywords: semilinear wave equation
zbMATH Keywords: memory
zbMATH Keywords: existence
zbMATH Keywords: uniqueness
zbMATH Keywords: exponential decay
author: M. M. Cavalcanti
author: V. N. Domingos Cavalcanti
author: J. A. Soriano
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 1839927