Global existence of solutions to semilinear damped wave equation with slowly decaying initial 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-\Delta u + \partial _tu = f(u) & \text{in}\ \Omega\times (0,T), \\ u=0 & \text{on}\ \partial \Omega\times (0,T), \\ u(0)=u_0, \partial _tu(0)=u_1 & \text{in}\ \Omega \end{cases} \end{equation*} in an exterior domain $\Omega$ in $\mathbb R^N$ $(N\geq 2)$, where $f:\mathbb R\to \mathbb R$ is a smooth function which behaves like $f(u)\sim |u|^p$. From the view point of weighted energy estimates given by Sobajima--Wakasugi [26], the existence of global-in-time solutions with small initial data in the sense of $\langle{x}\rangle^{\lambda}u_0, \langle{x}\rangle^{\lambda}\nabla u_0, \langle{x}\rangle^{\lambda}u_1\in L^2(\Omega)$ with $\lambda\in (0,\frac{N}{2})$ is shown under the condition $p\geq 1+\frac{4}{N+2\lambda}$. The lower and upper bounds for the lifespan of blowup solutions with small initial data $(\epsilon u_0,\epsilon u_1)$ are also given.