SEMILINEAR STOCHASTIC WAVE EQUATIONS

The existence and uniqueness of solutions to a class of semilinear stochastic hyperbolic equations in ℛ^d are considered. First an energy inequality for a linear stochastic hyperbolic equation is established. Then it is proven that there exists a unique continuous local solution for the associated nonlinear equation in the Sobolev space H₁(ℛ^d) when the nonlinear terms are locally bounded and Lipschitz-continuous. Under an additional condition on the energy bound, the solution exists for all time. The results are shown to be applicable to stochastic wave equations with polynomial nonlinearities of degree m with m≤3 for d=3, and for any m≥1 for d=1 or 2.