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November/December 2019 Global existence of solutions to semilinear damped wave equation with slowly decaying initial data in exterior domain
Motohiro Sobajima
Differential Integral Equations 32(11/12): 615-638 (November/December 2019).

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.

Citation

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Motohiro Sobajima. "Global existence of solutions to semilinear damped wave equation with slowly decaying initial data in exterior domain." Differential Integral Equations 32 (11/12) 615 - 638, November/December 2019.

Information

Published: November/December 2019
First available in Project Euclid: 22 October 2019

zbMATH: 07144906
MathSciNet: MR4021256

Subjects:
Primary: 35B33, 35L20

Rights: Copyright © 2019 Khayyam Publishing, Inc.

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Vol.32 • No. 11/12 • November/December 2019
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