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

We study the initial boundary value problem of an edge-degenerate wave equation. The operator $\Delta_{\mathbb{E}}$ with edge degeneracy on the boundary $\partial E$ 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 $E(0)\leq d$ and $I(u_0)>0$. Moreover, we obtain the result of blow-up with initial energy $E(0)\leq d$ and $I(u_0)