Global existence and uniform decay of solutions for a coupled system of nonlinear viscoelastic wave equations with not necessarily differentiable relaxation functions
type: Item
instance of: scholarly article
zbMATH Open document ID: 1250.35139
author: Wenjun Liu
author: Jun Yu
published in: Studies in Applied Mathematics
publication date: 2012-06-18T00:00:00Z
Mathematics Subject Classification ID: 35L53
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35R09
Mathematics Subject Classification ID: 35L77
zbMATH DE Number: 6047242
zbMATH Keywords: hyperbolic system
zbMATH Keywords: nonlinearity
zbMATH Keywords: global existence
zbMATH Keywords: decay of solutions
zbMATH Keywords: exponential decay
MaRDI profile type: Publication
full work available at URL: j.1467-9590.2011.00524.x
OpenAlex ID: W1531595286
cites work: Existence and uniform decay for a nonlinear viscoelastic equation with strong damping
cites work: Global existence and uniform stability of solutions for a quasilinear viscoelastic problem
cites work: Uniform stabilization of solutions of a nonlinear system of viscoelastic equations
cites work: Decay estimates for second order evolution equations with memory
cites work: REMARKS ON BLOW-UP AND NONEXISTENCE THEOREMS FOR NONLINEAR EVOLUTION EQUATIONS
cites work: General decay rate estimates for viscoelastic dissipative systems
cites work: Existence of a solution of the wave equation with nonlinear damping and source terms
cites work: Global existence and uniform decay for a nonlinear viscoelastic equation with damping
cites work: Decay estimates for some semilinear damped hyperbolic problems
cites work: Instability and Nonexistence of Global Solutions to Nonlinear Wave Equations of the Form Pu tt = -Au + ℱ(u)
cites work: Global nonexistence theorems for quasilinear evolution equations with dissipation
cites work: The exponential stabilization of the higher-dimensional linear system of thermoviscoelasticity
cites work: Blow up and global existence in a nonlinear viscoelastic wave equation
cites work: Global nonexistence theorems for a class of evolution equations with dissipation
cites work: Q4530284
cites work: Exponential Decay for The Semilinear Wave Equation with Locally Distributed Damping
cites work: Frictional versus Viscoelastic Damping in a Semilinear Wave Equation
cites work: Asymptotic Stability and Energy Decay Rates for Solutions of the Wave Equation with Memory in a Star-Shaped Domain
cites work: Q4440778
cites work: General decay and blow-up of solution for a quasilinear viscoelastic problem with nonlinear source
cites work: Asymptotic behavior for a nonlinear viscoelastic problem with a velocity-dependent material density
cites work: General decay rate estimate for a viscoelastic equation with weakly nonlinear time-dependent dissipation and source terms
cites work: A new method to obtain decay rate estimates for dissipative systems with localized damping
cites work: The global Cauchy problem for a relativistic scalar field with power interaction
cites work: Weak solutions for a system of nonlinear Klein-Gordon equations
cites work: On the standing wave in coupled non-linear Klein-Gordon equations
cites work: Q5708842
cites work: Polynomial stability without polynomial decay of the relaxation function
cites work: Polynomial stability of solutions for a system of non-linear viscoelastic equations
cites work: Non-existence of global solutions for nonlinear strongly damped hyperbolic systems
cites work: Global and blow-up solutions for a system of nonlinear hyperbolic equations with dissipative terms
cites work: General decay of solutions of a nonlinear system of viscoelastic equations
cites work: Uniform decay of solutions for a quasilinear system of viscoelastic equations
cites work: On decay and blow-up of the solution for a viscoelastic wave equation with boundary damping and source terms
cites work: On existence, uniform decay rates and blow up for solutions of systems of nonlinear wave equations with damping and source terms
DOI: 10.1111/J.1467-9590.2011.00524.X
title: Global existence and uniform decay of solutions for a coupled system of nonlinear viscoelastic wave equations with not necessarily differentiable relaxation functions
review text: The authors deal with the initial-boundary value problem for the coupled system of nonlinear viscoelastic wave equations NEWLINE\[NEWLINE\begin{aligned} &|u_t|^\rho u_{tt}-\Delta u-\Delta u_{tt}+\int_0^t g(t-\tau)\Delta u(x,\tau)d\tau -\Delta u_t=f(u,v),\;(x,t)\in \Omega\times (0,\infty),\\ &|v_t|^\rho v_{tt}-\Delta v-\Delta v_{tt}+\int_0^t g(t-\tau)\Delta v(x,\tau)d\tau -\Delta v_t=k(u,v),\;(x,t)\in \Omega\times (0,\infty),\\ &u(x,t)=v(x,t)=0,\;(x,t)\in \partial \Omega\times (0,\infty),\\ &u(x,0)=u_0(x),\;u_t(x,0)=u_1(x),\;x\in \Omega,\end{aligned} NEWLINE\]NEWLINE where \(\Omega\) is a bounded domain in \(\mathbb{R}^n,\;n\geq 1;\) with a smooth boundary,\ \(\rho\) is a real number such that \(0<\rho<2/(n-2)\) if \(n\geq 3\) or \(\rho > 0\) if \(n=1,2.\) The purpose of the paper is to establish the global existence and exponential decay of solutions under weaker conditions on the relaxation functions that are not necessarily differentiable.
reviewed by: Igor Bock
description: scientific article; zbMATH DE number 6047242