The critical parameter for the heat equation with a noise term to blow up in finite time
Carl Mueller · The Annals of Probability · 2000
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 0. $$ It was known that if $\gamma3/2$, then there is a positive probability that u blows up in finite time.