2021 Volume 64 Issue 2 Pages 137-162
This paper is concerned with the blowup phenomena for the initial value problem of the wave equation with a critical space-dependent damping term V0|x|-1 and a p-th order power nonlinearity on the Euclidean space RN, where N ≥ 3 and V0 ∈ [0,(N-1)2/(N+1)). The main result of the present paper is to prove existence of a unique local solution of the problem and to provide a sharp estimate for lifespan for such a solution for small data with a compact support when N/(N-1) < p ≤ pS(N + V0), where pS(N) is the Strauss exponent for the semilinear wave equation without damping. The main idea of the proof is due to the technique of test functions for the classical wave equation originated by Zhou-Han (2014). Consequently, the result poses whether the value V0 = (N-1)2/(N + 1) is a threshold for diffusive structure of the singular damping |x|-1 or not.
