Blowup for the heat equation with a noise term

Carl Mueller, Richard B Sowers

Research output: Contribution to journalArticlepeer-review

Abstract

In this paper we study blow up of the equation {Mathematical expression}, where {Mathematical expression} is a two-dimensional white noise field and where Dirichlet boundary conditions are enforced. It is known that if γ<3/2, then the solution exists for all time; in this paper we show that if γ 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. Offspring are peaks which are higher by some factor. We show that the expected number of offspring is greater than one when γ is much larger than 3/2, and thus the branching process survives with positive probability, corresponding to blowup in finite time.

Original languageEnglish (US)
Pages (from-to)287-320
Number of pages34
JournalProbability Theory and Related Fields
Volume97
Issue number3
DOIs
StatePublished - Sep 1 1993
Externally publishedYes

Keywords

  • Mathematics Subject Classification (1985): 60H15

ASJC Scopus subject areas

  • Analysis
  • Statistics and Probability
  • Statistics, Probability and Uncertainty

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