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

Abstract

Consider the stochastic partial differential equation ut = uxx + uγẆ, where x ∈ I ≡ [0, J], Ẇ = Ẇ(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 γ < 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 γ sufficiently large. We show that if γ > 3/2, then there is a positive probability that u blows up in finite time.