# Blow-up for the one dimensional stochastic wave equations

## TL;DR

This paper investigates conditions under which solutions to a class of one-dimensional stochastic wave equations become unbounded in finite time, addressing an open problem and introducing a new comparative approach.

## Contribution

It develops an $ ext{Omega}_ ext{delta}$-comparative method to prove finite-time blow-up of solutions under specific nonlinear noise conditions.

## Key findings

- Solutions blow up in finite time with positive probability.
- The approach addresses an open problem in stochastic wave equations.
- Conditions on initial data and noise are critical for blow-up.

## Abstract

The paper is concerned with the problem of explosive solutions for a class of semilinear stochastic wave equations. The challenging open problem(\cite{CMullR}) which is raised by C.Mueller and G.Richards is included in this problem.We develop an $\Omega_\delta$-comparative approach. With the aid of new approach, under appropriate conditions on the initial data and the nonlinear multiplicative noise term $(c_2u+f(u)) \dot{W}(t,x)$ with $|f(u)|\geq \kappa |u|^r,r>1,\kappa>0$, we prove in Theorem 3.1 that the solutions to the stochastic wave equation will blow up in finite time with positive probability.