Decay rates of solutions of an anisotropic inhomogeneousn-dimensional viscoelastic equation with polynomially decaying kernels

Abstract

We consider the anisotropic and inhomogeneous viscoelastic equation and we prove that the first and second order energy decay polynomially as time goes to infinity when the relaxation function also decays polynomially to zero. That is, if the kernelG _ijkl satisfies

$$\dot G_{ijkl} \leqq - c_0 G_{ijkl}^{1 + \frac{1}{p}} ;and G_{ijkl} ,G_{ijkl}^{1 + \frac{1}{p}} \in L^1 (\mathbb{R})for p > 2such that 2^m - 1< p,$$

then the first and second order energy decay as$\frac{1}{{(1 + t)^q }}$ withq=2 ^m−1.