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 𝜕_𝑡² 𝑢 −Δ𝑢 + \frac{𝑉₀}{|𝑥|} 𝜕_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 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.