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.

中文翻译：

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外部域中具有初始衰减的半线性阻尼波方程解的整体存在

在本文中，我们讨论半线性阻尼波方程\ begin {equation *} \ begin {cases} \ partial _t ^ 2u- \ Delta u + \ partial _tu = f（u）＆\ text的弱解的整体存在性{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 *}在外部域$ \ Omega $在$ \ mathbb R ^ N $ $（N \ geq 2）$中，其中$ f：\ mathbb R \至\ mathbb R $是一个平滑函数，其行为类似于$ f（u）\ sim | u | ^ p $。从Sobajima--Wakasugi [26]给出的加权能量估计的角度来看，存在以$ \ langle {x} \ rangle ^ {\ lambda} u_0表示的，初始数据较小的全局解决方案， \ langle {x} \ rangle ^ {\ lambda} \ nabla u_0，\ langle {x} \ rangle ^ {\ lambda} u_1 \ in L ^ 2（\ Omega）$和$ \ lambda \ in（0，在条件$ p \ geq 1+ \ frac {4} {N + 2 \ lambda} $下显示\ frac {N} {2}）$。还给出了具有较小初始数据$（\ epsilon u_0，\ epsilon u_1）$的爆炸解决方案寿命的上限和下限。