On global existence for semilinear wave equations with spacedependent critical damping

TL;DR

This paper investigates the conditions for global existence and blowup of solutions to semilinear wave equations with space-dependent critical damping, providing new energy estimates and clarifying initial data support effects.

Contribution

It introduces a weighted energy estimate for the linear problem that clarifies the influence of initial data support on global existence for these wave equations.

Findings

Weighted energy estimate established, independent of initial data support location

Conditions for global existence and blowup are characterized

Alternative proof of existing energy estimates provided

Abstract

The global existence for semilinear wave equations with space-dependent critical damping $\partial_t^2u-\Delta u+\frac{V_0}{|x|}\partial_t u=f(u)$ in an exterior domain is dealt with, where $f(u)=|u|^{p-1}u$ and $f(u)=|u|^p$ are in mind. Existence and non-existence of global-in-time solutions are discussed. To obtain global existence, a weighted energy estimate for the linear problem is crucial. The proof of such a weighted energy estimate contains an alternative proof of energy estimates established by Ikehata--Todorova--Yordanov [J.\ Math.\ Soc.\ Japan (2013), 183--236] but this clarifies the precise independence of the location of the support of initial data. The blowup phenomena is verified by using a test function method with positive harmonic functions satisfying the Dirichlet boundary condition.