Asymptotic stability and blow-up of solutions for an edge-degenerate wave equation with singular potentials and several nonlinear source terms of different sign

Abstract

We study the initial boundary value problem of an edge-degenerate wave equation. The operator ΔE with edge degeneracy on the boundary ∂E was investigated in the literature. We give the invariant sets and the vacuum isolating behavior of solutions by introducing a family of potential wells. We prove that the solution is global in time and exponentially decays when the initial energy satisfies E(0) ≤ d and Ι(u0) > 0. Moreover, we obtain the result of blow-up with initial energy E(0) ≤ d and I(u0) < 0, and give a lower bound for the blow-up time T*.