Nonlinear stochastic wave equations: Blow-up of second moments in $L^2$-norm

TL;DR

This paper investigates conditions under which solutions to certain nonlinear stochastic wave equations in low-dimensional domains exhibit finite-time blow-up of their second moments in the mean-square sense, highlighting the impact of initial data, nonlinearity, and noise.

Contribution

It establishes a rigorous criterion for finite-time blow-up of the $L^2$-norm in stochastic wave equations, extending understanding of explosive solutions in stochastic PDEs.

Findings

Second moments blow up in finite time under specified conditions.

Theoretical results are supported by a concrete example.

Blow-up occurs in the mean-square sense for solutions with certain initial data and noise parameters.

Abstract

The paper is concerned with the problem of explosive solutions for a class of nonlinear stochastic wave equations in a domain $\mathcal{D}\subset\mathbb{R}^d$ for $d\leq3$. Under appropriate conditions on the initial data, the nonlinear term and the noise intensity is proved in Theorem 3.1 that the $L^2$-norm of the solution will blow up at a finite time in the mean-square sense. An example is given to show an application of the theorem.