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

Consider the stochastic partial differential equation ut = uxx + uγW, where x E I ≡ [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 < ∞ such that P(lim sup u(t, x) = ∞) > 0./t↑T x It was known that if y < 3/2, then with probability 1, u does not blow up in finite time. It was also known that there is a positive probability of finite time blowup for y sufficiently large. We show that if y > 3/2, then there is a positive probability that u blows up in finite time.