On global existence for semilinear wave equations with space-dependent critical damping

Abstract

The global existence for semilinear wave equations with space-dependent critical damping ∂ t u−∆u+ V0 |x| ∂tu = f(u) in an exterior domain is dealt with, where f(u) = |u|p−1u and f(u) = |u| 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. Mathematics Subject Classification (2010): Primary:35L71, 35A01, Secondary:35L20, 35B40,