Explosion of solutions to nonlinear stochastic wave equations with multiplicative noise

Let D⊂RdD⊂Rd be an open bounded domain in the dd-dimensional Euclidian space RdRd with smooth boundary ∂D∂D. In this paper we investigate the local existence, global existence and explosion of solutions to the following stochastic wave equation:equation(0.1){dX(t)=Y(t)dt,dY(t)=(ΔX(t)−|Y(t)|pY(t)+|X(t)|qX(t))dt+B(t,X(t),Y(t))dW(t),p,q>0,X(0)=X0,Y(0)=Y0,X(t)∣∂D=0, where Δ=∑∂2/∂xi2 is a Laplace operator. The process W(t)W(t) denotes a cylindrical Brownian process on a complete probability space (Ω,F,P)(Ω,F,P) with filtration of the usual condition.