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

We discuss an initial boundary value problem of stochastic viscoelastic wave equation driven by multiplicative noise involving the nonlinear damping term $|u_t|^{q-2}u_t$ and a source term of the type $|u|^{p-2}u$. We firstly 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 $q\geq p$. 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 $p>q$.