Non-blowup of stochastic heat equations by noise

RESUMO

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 $b$ will result in a blowup of the solution $u$ at a finite time. In this paper, we show that a random noise will prevent the explosion of the solution. More precisely, we show 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+u(t,x)^n\,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 global solution if \begin{equation} z\,b(z)\leq C_b(1+z^2)+\eta |z|^{2n}, \qquad z\in\R. \end{equation} for some $\eta\in [0, \frac{1}{2})$, here $n$ is an arbitrary, but fixed integer, $B_t, t\geq 0$ is a Brownian motion. Truncation and comparison techniques play an important role.