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

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,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ u_{tt}(t,x)-\Delta u(t,x)+ \int _{0}^{t} g(t-s)\Delta u(s,x)\,ds+f(u)= \sigma (u)\frac{\partial W(x,t)}{\partial t}, \quad x\in D\subset \mathbb{R}^{n}, $$\end{document} is considered, where f is a continuous function with polynomial growth of order less than or equal to n/(n−2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$n/(n-2)$\end{document} and σ is Lipschitz with σ(0)=0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\sigma (0)=0$\end{document}. By Tartar’s energy method, we prove that for any solution to the equation the propagate speed is finite.