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 $(c_{2}u+f(u))\dot{W}(t,x)$ with $|f(u)|\ge \kappa |u{|}^r,r>1,\kappa >0$, we prove in Theorem 3.1 that the solutions to the stochastic wave equation will blow up in finite time with positive probability.