Large time behaivor of global solutions to nonlinear wave equations with frictional and viscoelastic damping terms

TL;DR

This paper investigates the long-term behavior of solutions to nonlinear wave equations with frictional and viscoelastic damping, showing that frictional damping dominates and solutions behave like heat kernels for small initial data.

Contribution

It demonstrates that frictional damping remains dominant in the nonlinear case and establishes diffusion phenomena approximating solutions by heat kernels.

Findings

Solutions are approximated by heat kernels for small initial data.

Frictional damping dominates over viscoelastic damping in the nonlinear setting.

The approach is based on estimates for linear equations.

Abstract

In this paper, we study the Cauchy problem for a nonlinear wave equation with frictional and viscoelastic damping terms. As is pointed out by [8], in this combination, the frictional damping term is dominant for the viscoelastic one for the global dynamics of the linear equation. In this note we observe that if the initial data is small, the frictional damping term is again dominant even in the nonlinear equation case. In other words, our main result is diffusion phenomena: the solution is approximated by the heat kernel with a suitable constant. Our proof is based on several estimates for the corresponding linear equations.