Global existence of solutions to semilinear damped wave equation with slowly decaying inital data in exterior domain

Abstract

In this paper, we discuss the global existence of weak solutions to the semilinear damped wave equation \begin{equation*} \begin{cases} \partial_t^2u-Δu + \partial_tu = f(u) & \text{in}\ Ω\times (0,T), \\ u=0 & \text{on}\ \partialΩ\times (0,T), \\ u(0)=u_0, \partial_tu(0)=u_1 & \text{in}\ Ω, \end{cases} \end{equation*} in an exterior domain $\Omega$ in ${\mathbb{R}}^N$ $(N\ge 2)$, where $f:\mathbb{R}\to \mathbb{R}$ is a smooth function behaves like $f(u)\sim |u{|}^p$. From the view point of weighted energy estimates given by Sobajima--Wakasugi \cite{SoWa4}, the existence of global-in-time solutions with small initial data in the sense of $(1+|x{|}^2{)}^{\lambda /2}{u}_0$, $(1+|x{|}^2{)}^{\lambda /2}\nabla u_0$, $(1+|x{|}^2{)}^{\lambda /2}{u}_1\in L^2(\Omega )$ with $\lambda \in (0,\frac{N}{2})$ is shown under the condition $p\ge 1+\frac{4}{N+2\lambda }$. The sharp lower bound for the lifespan of blowup solutions with small initial data $(\epsilon u_0,\epsilon u_1)$ is also given.

16 pages