Stochastic Wave Equations with Nonlinear Damping and Source Terms

TL;DR

This paper investigates stochastic wave equations with nonlinear damping and source terms, establishing local and global existence of solutions, and analyzing conditions leading to solution blow-up or explosion.

Contribution

It introduces a rigorous analysis of existence, uniqueness, and blow-up phenomena for stochastic wave equations with nonlinear damping and source terms.

Findings

Global solutions for q ≥ p

Local solutions may blow up for p > q

Energy explosion occurs with positive probability

Abstract

In this paper, we discuss an initial boundary value problem for the stochastic wave equation involving the nonlinear damping term $|u_t|^{q-2}u_t$ and a source term of the type $|u|^{p-2}u$. We firstly establish the local existence and uniqueness of solution by the Galerkin approximation method and show that the solution is global for $q\geq p$. Secondly, by an appropriate energy inequality, the local solution of the stochastic equations will blow up with positive probability or explosive in energy sense for $p>q$.