Smoothing effect and asymptotic behavior of solutions to nonlinear elastic wave equations with viscoelastic terms in the framework of $L^{p}$-Sobolev spaces

Abstract

The Cauchy problem for nonlinear elastic wave equations with viscoelastic damping terms is investigated in $L^{p}$ framework. It is proved that the small global solutions constructed in $L^{2}$-Sobolev spaces in our preceding paper [12] satisfies consistency property corresponding to the additional regularity of the initial data. As a result, sharp estimates in $t$ and approximation formulas by the diffusion waves are established.