Non-blowup of stochastic heat equations by noise

Abstract

Consider the heat equation: \begin{equation}\label{eq:00} \begin{cases} \displaystyle du(t,x)=\frac12\Delta u(t,x)\,dt+b(u(t,x))\,dt,&t>0,\ x\in D,\\[1mm] u(t,x)=0,&t\ge0,\ x\in\partial D, u(0,x)=u_0(x),&x\in D. \end{cases} \end{equation} It is well known that superlinear growth of the coefficient $b$ may lead to finite-time blow-up. In this paper, we show that, for every locally Lipschitz reaction coefficient $b:\R\to\R$, there always exists a locally Lipschitz noise coefficient $\sigma:\R\to\R$ such that the stochastic heat equation \begin{equation}\label{eq:01} \begin{cases} \displaystyle du(t,x)=\frac12\Delta u(t,x)\,dt+b(u(t,x))\,dt+\sigma(u(t,x))\,dB_t,&t>0,\ x\in D,\\[1mm] u(t,x)=0,&t\ge0,\ x\in\partial D, u(0,x)=u_0(x),&x\in D. \end{cases} \end{equation} has a unique global solution for every $u_0\in C_c(D)$. Here $(B_t)_{t\ge0}$ is a real-valued Brownian motion. In particular, if $b=0$, the equation has a global solution for every locally Lipschitz $\sigma$, without any growth restriction at infinity. The proof uses reflected scalar barriers, a Lyapunov estimate, truncation, and comparison techniques.