Blow-up for the one dimensional stochastic wave equations

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.