EXPONENTIAL DECAY FOR A NONLINEAR VISCOELASTIC EQUATION WITH SINGULAR KERNELS

doi:10.1016/S0252-9602(12)60173-8

Abstract

Abstract:

The nonlinear viscoelastic wave equation
|u_t|^ρ u_tt − Δu − Δu_tt +∫^t₀g(t − s)Δu(s)ds + |u|^p u = 0,
in a bounded domain with initial conditions and Dirichlet boundary conditions is consid-ered. We prove that, for a class of kernels g which is singular at zero, the exponential decay rate of the solution energy. The result is obtained by introducing an appropriate Lyapounov functional and using energy method similar to the work of Tatar in 2009. This work improves earlier results.

Key words: viscoelastic wave equation, kernel, exponential decay, memory term, singu-lar kernel

CLC Number:

35B40

Shun-Tang Wu. EXPONENTIAL DECAY FOR A NONLINEAR VISCOELASTIC EQUATION WITH SINGULAR KERNELS[J].Acta mathematica scientia,Series B, 2012, 32(6): 2237-2246.

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