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Blowup for the heat equation with a noise term
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  • Published: September 1993

Blowup for the heat equation with a noise term

  • Carl Mueller1 &
  • Richard Sowers2 

Probability Theory and Related Fields volume 97, pages 287–320 (1993)Cite this article

  • 297 Accesses

  • 25 Citations

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Summary

In this paper 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 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 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 γ is much larger than 3/2, and thus the branching process survives with positive probability, corresponding to blowup in finite time.

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Author information

Authors and Affiliations

  1. Department of Mathematics, University of Rochester, 14627, Rochester, NY, USA

    Carl Mueller

  2. Center for Applied Mathematical Sciences, University of Southern California, 90089-1113, Los Angeles, CA, USA

    Richard Sowers

Authors
  1. Carl Mueller
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  2. Richard Sowers
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Additional information

Supported by NSF grant DMS-9021508, NSA grant MDA904-910-H-0034, and ARO Grant MSI DAAL03-91-C-0027

Supported by ONR grant N00014-91-J-1526.

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Mueller, C., Sowers, R. Blowup for the heat equation with a noise term. Probab. Th. Rel. Fields 97, 287–320 (1993). https://doi.org/10.1007/BF01195068

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  • Received: 30 April 1992

  • Revised: 23 April 1993

  • Issue Date: September 1993

  • DOI: https://doi.org/10.1007/BF01195068

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Mathematics Subject Classification (1985)

  • 60H15
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