Variational inequality for quasilinear wave equations with nonlinear damping terms
type: Item
instance of: scholarly article
title: Variational inequality for quasilinear wave equations with nonlinear damping terms
zbMATH Open document ID: 1004.35100
DOI: 10.1016/S0362-546X(00)00239-X
author: Jeong Ja Bae
author: Jong Yeoul Park
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2002-08-25T00:00:00Z
review text: The paper deals with the unilateral problem \(u_{tt}-M(\|\nabla u\|^2_2+\|\nabla v\|^2_2)\Delta u+\delta |u_t|^p u_t\geq \mu |u|^\alpha u ,\) \(v_{tt}-M(\|\nabla u\|^2_2+\|\nabla v\|^2_2)\Delta v+\delta |v_t|^p v_t\geq \mu |v|^\alpha v\), \(u\geq 0\), \(v\geq 0,\) \((u_{tt}-M(\|\nabla u\|^2_2+\|\nabla v\|^2_2)\Delta u+\delta |u_t|^p u_t- \mu |u|^\alpha u)u = 0 ,\) \((v_{tt}-M(\|\nabla u\|^2_2+\|\nabla v\|^2_2)\Delta v+\delta |v_t|^p v_t- \mu |v|^\alpha v)v=0\), \(\text{in} Q,\) \(u=v=0 \;\text{in} \Sigma\), \(u(0)=u_0\geq 0\), \(v(0)=v_0\geq 0\), \(u_t(0)-u_1\geq 0\), \(v_t(0)-v_1\geq 0\), \((u_t(0)-u_1)u_0=0\), \((v_t(0)-v_1)v_0=0 \;\text{in} \Omega,\) where \(\Omega\) is a bounded domain in \(\mathbb R^N\) with smooth boundary \(\Gamma\), \(T>0\), \(Q=\Omega \times (0,T), \Sigma=\Gamma\times (0,T)\), \(M(s)=a+bs\), \(a+b\geq 0, b\geq 0\), \(\alpha>0\), \(\rho\geq 0\), \(\delta>0\), \(\mu\in \mathbb R.\) The authors formulate the penalized problem and employ the Galerkin method to construct its solution. They derive a priori estimates and obtain the existence and uniqueness of a penalized solution. The existence of a weak solution \(\{u,v\}\in L^\infty(0,T;H^1_0(\Omega)\cap H^2(\Omega))^2\), \(\{u',v'\}\in [L^\infty(0,T;L^2(\Omega))\cap L^2(0,T;H_0^1(\Omega))]^2\;\) of the original unilateral problem is obtained by performing the limit of the subsequence of the penalized solutions \(\{u_\varepsilon,v_\varepsilon\}\).
reviewed by: Igor Bock
Mathematics Subject Classification ID: 35L80
Mathematics Subject Classification ID: 49J40
zbMATH DE Number: 1788057
zbMATH Keywords: existence of a solution
zbMATH Keywords: penalization
zbMATH Keywords: Galerkin method
zbMATH Keywords: a priori estimates
zbMATH Keywords: unilateral problem
MaRDI profile type: Publication
cites work: Equivalent Norms for Sobolev Spaces
cites work: Local solutions for a nonlinear degenerate Hyperbolic equation
cites work: Global attractors for semilinear damped wave equations with supercritical exponent
cites work: On global existence and uniqueness for the unilateral problem associated to the degenerated Kirchhoff equation
cites work: On global solutions and energy decay for the wave equations of Kirchhoff type with nonlinear damping terms
cites work: Non-Linear vibration of an elastic string
cites work: Q4366334
cites work: Global existence, decay, and blowup of solutions for some mildly degenerate nonlinear Kirchhoff strings
cites work: Q4210891
cites work: Variational inequality for a nonlinear model of the oscillations of beams
cites work: Exponential decay for a nonlinear system of hyperbolic equations with locally distributed dampings
cites work: Infinite-dimensional dynamical systems in mechanics and physics
full work available at URL: s0362-546x(00)00239-x
OpenAlex ID: W1996020088
description: scientific article; zbMATH DE number 1788057