Global existence and explosion of the stochastic viscoelastic wave equation driven by multiplicative noises

Abstract

In this paper, we discuss an initial boundary value problem of stochastic viscoelastic wave equation driven by multiplicative noise involving the nonlinear damping term utq−2ut and a source term of the type up−2u. We first establish the local existence and uniqueness of solution by the iterative technique truncation function method. Moreover, we also show that the solution is global for q ≥ p. Lastly, by modifying the energy functional, we give sufficient conditions such that the local solution of the stochastic equations will blow up with positive probability or explode in energy sense for p > q.