General decay rate estimate for a viscoelastic equation with weakly nonlinear time-dependent dissipation and source terms

A viscoelastic wave equation in canonical form with weakly nonlinear time-dependent dissipation and source terms is investigated in this paper. For a wider class of relaxation functions and without imposing any restrictive growth assumption on the damping term at the origin, we establish an explicit and general energy decay rate result.

[1]  S. Messaoudi,et al.  On the control of solutions of viscoelastic equations with boundary feedback , 2009 .

[2]  Marcelo M. Cavalcanti,et al.  Existence and uniform decay rates for viscoelastic problems with nonlinear boundary damping , 2001, Differential and Integral Equations.

[3]  Enrique Zuazua,et al.  Exponential Decay for The Semilinear Wave Equation with Locally Distributed Damping , 1990 .

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

[5]  Xiaosen Han,et al.  General decay of energy for a viscoelastic equation with nonlinear damping , 2009, J. Frankl. Inst..

[6]  Enzo Vitillaro,et al.  Global Nonexistence Theorems for a Class of Evolution Equations with Dissipation , 1999 .

[7]  Howard A. Levine,et al.  Global Nonexistence Theorems for Quasilinear Evolution Equations with Dissipation , 1997 .

[8]  Vladimir Georgiev,et al.  Existence of a Solution of the Wave Equation with Nonlinear Damping and Source Terms , 1994 .

[9]  Wenjun Liu.  General decay and blow-up of solution for a quasilinear viscoelastic problem with nonlinear source , 2010 .

[10]  P. Martinez.  A new method to obtain decay rate estimates for dissipative systems with localized damping , 1999 .

[11]  Howard A. Levine,et al.  Global existence and global nonexistence of solutions of the Cauchy problem for a nonlinearly damped wave equation , 1998 .

[12]  Marcelo Moreira Cavalcanti,et al.  Frictional versus Viscoelastic Damping in a Semilinear Wave Equation , 2003, SIAM J. Control. Optim..

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

[14]  D. Sattinger,et al.  Saddle points and instability of nonlinear hyperbolic equations , 1975 .

[15]  Nasser-eddine Tatar,et al.  Global existence and uniform stability of solutions for a quasilinear viscoelastic problem , 2007 .

[16]  Xiaosen Han,et al.  Global existence and uniform decay for a nonlinear viscoelastic equation with damping , 2009 .

[17]  Mingxin Wang,et al.  Global and blow-up solutions for a system of nonlinear hyperbolic equations with dissipative terms☆ , 2006 .

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

[19]  Howard A. Levine,et al.  Some Additional Remarks on the Nonexistence of Global Solutions to Nonlinear Wave Equations , 1974 .

[20]  Weijiu Liu.  The exponential stabilization of the higher-dimensional linear system of thermoviscoelasticity , 1998 .

[21]  Jong Yeoul Park,et al.  Existence and asymptotic stability for the semilinear wave equation with boundary damping and source term , 2008 .

[22]  Graham H. Williams,et al.  Partial exact controllability for the linear thermo-viscoelastic model , 1998 .

[23]  S. Messaoudi.  General decay of the solution energy in a viscoelastic equation with a nonlinear source , 2008 .

[24]  Salim A. Messaoudi,et al.  Blow up and global existence in a nonlinear viscoelastic wave equation , 2003 .

[25]  Marcelo M. Cavalcanti,et al.  Existence and uniform decay for a non‐linear viscoelastic equation with strong damping , 2001 .

[26]  Jong Yeoul Park,et al.  Well-posedness and uniform decay rates for the Klein–Gordon equation with damping term and acoustic boundary conditions , 2009 .

[27]  Howard A. Levine,et al.  Instability and Nonexistence of Global Solutions to Nonlinear Wave Equations , 1974 .

[28]  Non-existence of global solutions for nonlinear strongly damped hyperbolic systems , 2005 .

[29]  Patrick Martinez,et al.  Precise decay rate estimates for time-dependent dissipative systems , 2000 .