Global existence of solutions to semilinear damped wave equation with slowly decaying initial data in exterior domain
Global existence of solutions to semilinear damped wave equation with slowly decaying initial data in exterior domain
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DOI：
10.57262/die/1571731512
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发表时间：
2018-12
影响因子：
1.4
通讯作者：
M. Sobajima
中科院分区：
文献类型：
--
作者：
M. Sobajima
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.
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.