Blowup for the heat equation with a noise term
We study blow up of the equation \(u_ t=u_{xx}+u^ \gamma \dot W_{tx}\), where \(\dot W_{tx}\) is a two-dimensional white noise field and where Dirichlet boundary conditions are enforced. It is known that if \(\gamma<3/2\), then the solution exists for all time; in this paper we show that if \(\gamma\) 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 of \(u\) to a branching process. Large peaks are regarded as particles in this branching process. Offsprings are peaks which are higher by some factor. We show that the expected number of offsprings is greater than one when \(\gamma\) is much larger than 3/2, and thus the branching process survives with positive probability, corresponding to blowup in finite time.
scientific article; zbMATH DE number 569004