Non-existence results for stochastic wave equations in one dimension

The purpose of this paper is to extend recent results of [2] and [10] for the stochastic heat equation to the stochastic wave equation given by|∂2u(t,x)∂t2=∂2u(t,x)∂x2+σ(u(t,x))W˙(t,x)+b(u(t,x)),x∈D∖∂D,t>0,u(0,x)=u0(x),∂∂tu(0,x)=v0(x),x∈D, where W˙ is space-time white noise, σ is a real-valued globally Lipschitz function but b is assumed to be only locally Lipschitz continuous. Three types of domain conditions are studied: D=[0,1] with homogeneous Dirichlet boundary conditions, D=[0,2π] with periodic boundary conditions, and D=R. Then, under suitable conditions, the following integrability condition∫α∞1[β2+2∫αsb(r)dr]1/2ds<∞, for some α>0 and β>0, is studied in relation to non-existence of global solutions.