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

TL;DR

This paper studies the long-term behavior and smoothing effects of solutions to nonlinear elastic wave equations with viscoelastic damping, establishing sharp decay estimates and diffusion wave approximations in an $L^{p}$ setting.

Contribution

It extends previous $L^{2}$ results to $L^{p}$ spaces, proving consistency, smoothing effects, and diffusion wave approximations for solutions.

Findings

Solutions exhibit smoothing effects and asymptotic diffusion wave behavior.

Sharp decay estimates in $L^{p}$ norms are established.

Consistency with initial data regularity is confirmed.

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.