Smoothing effect and large time behavior of solutions to nonlinear elastic wave equations with viscoelastic term

TL;DR

This paper studies the decay, smoothing, and long-term behavior of solutions to 3D nonlinear elastic wave equations with viscoelastic damping, revealing their asymptotic profiles under small initial data.

Contribution

It provides new insights into the decay rates, smoothing effects, and asymptotic profiles of solutions to nonlinear elastic wave equations with viscoelastic damping.

Findings

Solutions exhibit decay and smoothing properties over time.

Asymptotic profiles of solutions are characterized as time approaches infinity.

Results are valid for small initial data in three-dimensional space.

Abstract

The Cauchy problem for a nonlinear elastic wave equations with viscoelastic damping terms is considered on the 3 dimensional whole space. Decay and smoothing properties of the solutions are investigated when the initial data are sufficiently small; and asymptotic profiles as $t \to \infty$ are also derived.