On the exponential decay of waves with memory

In this paper we consider general linear damped wave equations with memory. We establish energy estimates that under the assumption of exponentially bounded kernels, induce exponential decaying solutions. Numerical waves that mimic their continuous counterpart are also introduced using a finite element approach.

[1]  O. Staffans.  On a Nonlinear Hyperbolic Volterra Equation , 1980 .

[2]  Michael Renardy,et al.  Mathematical problems in viscoelasticity , 1987 .

[3]  S. Messaoudi,et al.  A general stability result for a quasilinear wave equation with memory , 2013 .

[4]  J. R. Branco,et al.  A singular perturbation of the heat equation with memory , 2008 .

[5]  Vidar Thomée,et al.  Long-time numerical solution of a parabolic equation with memory , 1994 .

[6]  Laurent Simon,et al.  Molecular Transport in Viscoelastic Materials: Mechanistic Properties and Chemical Affinities , 2014, SIAM J. Appl. Math..

[7]  Jong Yeoul Park,et al.  Decay rate estimates for wave equations of memory type with acoustic boundary conditions , 2011 .

[8]  N. Tatar.  Arbitrary decays in linear viscoelasticity , 2011 .

[9]  Mauro de Lima Santos,et al.  Decay rates for solutions of semilinear wave equations with a memory condition at the boundary. , 2002 .

[10]  Mauro de Lima Santos,et al.  Decay rates for solutions of a system of wave equations with memory , 2002 .

[11]  F. Hirosawa.  Energy Decay for a Degenerate Hyperbolic Equation with a Dissipative Term , 1999 .

[12]  A. Morro,et al.  Mathematical problems in linear viscoelasticity , 1987 .

[13]  Patrick Martinez,et al.  General decay rate estimates for viscoelastic dissipative systems , 2008 .

[14]  J. H. Liu,et al.  Singular perturbations of integrodifferential equations in Banach space , 1994 .

[15]  Carlos Lizama,et al.  Singular perturbations of integro-differential equations , 2006, Appl. Math. Comput..

[16]  J. Nohel.  A nonlinear hyperbolic volterra equation , 1979 .

[17]  J. A. Soriano,et al.  Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping , 2002 .

[18]  M. Santos,et al.  Lack of exponential decay of a coupled system of wave equations with memory , 2011 .

[19]  A decay result to a viscoelastic problem in Rn with an oscillating kernel , 2010 .

[20]  Jin Liang,et al.  Hyperbolic singular perturbations for integrodifferential equations , 2005, Appl. Math. Comput..

[21]  James H. Liu,et al.  Convergence for Hyperbolic Singular Perturbation of Integrodifferential Equations , 2007 .