On global existence for semilinear wave equations with spacedependent critical damping

Abstract

The global existence for semilinear wave equations with space-dependent critical damping ${\partial }_t^{2}u-\Delta u+\frac{V_0}{|x|}{\partial }_{t}u=f(u)$ 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.

21 pages