  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 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 $p\geq 1+\frac{4}{N+2\lambda}$. The sharp lower bound
for the lifespan of blowup solutions with small initial data $(\varepsilon
u_0,\varepsilon u_1)$ is also given.
