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October 2000 The critical parameter for the heat equation with a noise term to blow up in finite time
Carl Mueller
Ann. Probab. 28(4): 1735-1746 (October 2000). DOI: 10.1214/aop/1019160505

Abstract

Consider the stochastic partial differential equation $$u_t=u_{xx}+u^{\gamma W},$$ where $x\in\mathbf{I}\equiv[0,J], W=W(t,x)$ 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 $T < \infty$ such that $$ P(\lim_{t \uparrow T} \sup_{x} u(t, x) = \infty) > 0. $$ It was known that if $\gamma<3/2$, then with probability 1, $u$ does not blowup in finite time. It was also known that there is a positive probability of finite time blowup for $\gamma$ sufficiently large.

We show that if $\gamma>3/2$, then there is a positive probability that u blows up in finite time.

Citation

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Carl Mueller. "The critical parameter for the heat equation with a noise term to blow up in finite time." Ann. Probab. 28 (4) 1735 - 1746, October 2000. https://doi.org/10.1214/aop/1019160505

Information

Published: October 2000
First available in Project Euclid: 18 April 2002

zbMATH: 1044.60051
MathSciNet: MR1813841
Digital Object Identifier: 10.1214/aop/1019160505

Subjects:
Primary: 60H15
Secondary: 35L05 , 35R60

Keywords: heat equation , Stochastic partial differential equations , White noise

Rights: Copyright © 2000 Institute of Mathematical Statistics

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Vol.28 • No. 4 • October 2000
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