Well-posedness for the semilinear wave equations with nonlinear damping on manifolds with conical degeneration

TL;DR

This paper investigates the well-posedness, energy decay, and blow-up phenomena of semilinear wave equations with nonlinear damping and source terms on manifolds with conical singularities, providing conditions for global existence and finite-time blow-up.

Contribution

It establishes local and global existence, energy decay, and blow-up criteria for semilinear wave equations on manifolds with conical degeneration, using semigroup methods and potential well techniques.

Findings

Proved local existence and uniqueness of solutions.

Established energy decay and conditions for global existence.

Identified criteria for finite-time blow-up of solutions.

Abstract

This paper deals with a class of semilinear wave equation with nonlinear damping term $|u_{t}|^{m-2}u_t $ and nonlinear source term $g(x)|u|^{p-2}u$ on the manifolds with conical singularities. Firstly, we prove the local existence and uniqueness of the solution by the semigroup method. Secondly, we establish the global existence, the energy decay estimate and the blow-up in finite time of the solution with subcritial ($E(0)<d$) and critial ($E(0)=d$) initial energy level by constructing potential wells. We also show that the solution is global provided the damping dominates the source (that is $m\geq p$). Moreover, we prove the blow-up in finite time of the solution with arbitrary positive initial energy and give some necessary and sufficient condition for existing finite time blow-up solutions.