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

Abstract

In this paper, we study the Cauchy problem for a nonlinear wave equation with frictional and viscoelastic damping terms in ${\mathbb R}^{n}$. As is pointed out by [10], 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. Especially, the result obtained for the $n = 3$ case is essentially new. Our proof is based on several estimates for the corresponding linear equations.