On global existence for semilinear wave equations with spacedependent critical damping

Abstract

The global existence for semilinear wave equations with space-dependent critical dampingin 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.