The critical parameter for the heat equation with a noise term to blow up in finite time

Abstract

Consider the stochastic partial differential equation $$u_t=u_{xx}+u^{\gamma W},$$ where $x\in\mathbf{I}\equiv[0,J], W=W(t,x)$ is 2-parameter white noise, and we assume that the initial function $u(0,x)$ is nonnegative and not identically 0. We impose Dirichlet boundary conditions on u in the interval I. We say that u blows up in finite time, with positive probability, if there is a random time $T < \infty$ such that $$ P(\lim_{t \uparrow T} \sup_{x} u(t, x) = \infty) > 0. $$ It was known that if $\gamma<3/2$, then with probability 1, $u$ does not blowup in finite time. It was also known that there is a positive probability of finite time blowup for $\gamma$ sufficiently large. We show that if $\gamma>3/2$, then there is a positive probability that u blows up in finite time.