The finite speed of propagation for solutions to stochastic viscoelastic wave equation

Abstract

Abstract In this paper, a class of second order stochastic evolution equations with memory utt(t,x)−Δu(t,x)+∫0tg(t−s)Δu(s,x)ds+f(u)=σ(u)∂W(x,t)∂t,x∈D⊂Rn, $$ u_{tt}(t,x)-\Delta u(t,x)+ \int _{0}^{t} g(t-s)\Delta u(s,x)\,ds+f(u)= \sigma (u)\frac{\partial W(x,t)}{\partial t}, \quad x\in D\subset \mathbb{R}^{n}, $$ is considered, where f is a continuous function with polynomial growth of order less than or equal to n/(n−2) $n/(n-2)$ and σ is Lipschitz with σ(0)=0 $\sigma (0)=0$. By Tartar’s energy method, we prove that for any solution to the equation the propagate speed is finite.

Keywords

• Stochastic evolution equations
• The finite speed of propagation
• Multiplicative noise