Blowup for the heat equation with a noise term

Summary In this paper we study blow up of the equation , where is a two-dimensional white noise field and where Dirichlet boundary conditions are enforced. It is known that if <3/2, then the solution exists for all time; in this paper we show that if is much larger than 3/2, then the solution blows up in finite time with positive probability. We prove this by considering how peaks in the solution propagate. If a peak of high mass forms, we rescale the equation and divide the mass of the peak into a collection of peaks of smaller mass, and these peaks evolve almost independently. In this way we compare the evolution ofu to a branching process. Large peaks are regarded as particles in this branching process. Offspring are peaks which are higher by some factor. We show that the expected number of offspring is greater than one when is much larger than 3/2, and thus the branching process survives with positive probability, corresponding to blowup in finite time.Supported by NSF grant DMS-9021508, NSA grant MDA904-910-H-0034, and ARO Grant MSI DAAL03-91-C-0027Supported by ONR grant N00014-91-J-1526.