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

The global existence for semilinear wave equations with space-dependent critical damping ∂2 t u - Δu + V0 jxj ∂tu = f(u) in an exterior domain is dealt with, where f(u) = |u|p1u 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 the argument in this paper clarifies the precise dependence of the location of the support of initial data. The blowup phenomena are verified by using a test function method with positive harmonic functions satisfying the Dirichlet boundary condition.