Stochastic quasilinear viscoelastic wave equation with nonlinear damping and source terms

Abstract

Abstract The goal of this study is to investigate an initial boundary value problem for the stochastic quasilinear viscoelastic wave equation involving the nonlinear damping |ut|q−2ut $\vert u_{t} \vert ^{q-2} u_{t}$ and a source term of the type |u|p−2u $\vert u \vert ^{p-2}u$ driven by additive noise. By an appropriate energy inequality, we prove that finite time blow-up is possible for equation (1.1) below if p>{q,ρ+2} $p > \{q, \rho +2 \}$ and the initial data are large enough (that is, if the initial energy is sufficiently negative). Also, we show that if q≥p $q \geq p$, the local solution can be extended for all time and is thus global.

Keywords

• stochastic quasilinear viscoelastic wave equation
• blow-up of solution
• global existence