On potential wells to a semilinear hyperbolic equation with damping and conical singularity

Abstract

This article deals with a semilinear hyperbolic equation with damping and conical singularity that was built by Alimohammady et al. (2017) [1], where the weak solution with low initial energy (I(0)<d,d is the potential well depth) was considered. We extend the previous results in the following three aspects. Firstly, we consider the vacuum isolating behavior of the solution for initial energy I(0)≤0 and 0<I(0)<d, respectively. We find that there are two explicit vacuum regions: an annulus and a ball. Moreover, we obtain the asymptotic behavior of the energy functional as t tends to the maximal existence time, and then two necessary and sufficient conditions for the weak solution existing globally and blowing up in finite time are obtained. Secondly, we discuss the weak solution with critical initial energy and establish the global existence and nonexistence results. Finally, the weak solution with arbitrary positive initial energy is studied. In this case, the initial conditions such that the weak solution exists globally and blows up in finite time are given.