Asymptotic profile of solutions for semilinear wave equations with structural damping

Abstract

This paper is concerned with the initial value problem for semilinear wave equation with structural damping $u_{tt}+(-\Delta {)}^{\sigma }{u}_t-\Delta u=f(u)$, where $\sigma \in (0,\frac{1}{2})$ and $f(u)\sim |u{|}^p$ or $u|u{|}^{p-1}$ with $p>1+2/(n-2\sigma )$. We first show the global existence for initial data small in some weighted Sobolev spaces on ${\mathcal{R}}^n$ ($n\ge 2$). Next, we show that the asymptotic profile of the solution above is given by a constant multiple of the fundamental solution of the corresponding parabolic equation, provided the initial data belong to weighted $L^1$ spaces.