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Non-linear noise excitation and intermittency under high disorder


Authors: Davar Khoshnevisan and Kunwoo Kim
Journal: Proc. Amer. Math. Soc. 143 (2015), 4073-4083
MSC (2010): Primary 60H15, 60H25; Secondary 35R60, 60K37
DOI: https://doi.org/10.1090/S0002-9939-2015-12517-8
Published electronically: February 25, 2015
MathSciNet review: 3359595
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Abstract: Consider the semilinear heat equation $ \partial _t u = \partial ^2_x u + \lambda \sigma (u)\xi $ on the interval $ [0\,,L]$ with Dirichlet zero-boundary condition and a nice non-random initial function, where the forcing $ \xi $ is space-time white noise and $ \lambda >0$ denotes the level of the noise. We show that, when the solution is intermittent [that is, when $ \inf _z\vert\sigma (z)/z\vert>0$], the expected $ L^2$-energy of the solution grows at least as $ \exp \{c\lambda ^2\}$ and at most as $ \exp \{c\lambda ^4\}$ as $ \lambda \to \infty $. In the case that the Dirichlet boundary condition is replaced by a Neumann boundary condition, we prove that the $ L^2$-energy of the solution is in fact of sharp exponential order $ \exp \{c\lambda ^4\}$. We show also that, for a large family of one-dimensional randomly forced wave equations on $ \mathbf {R}$, the energy of the solution grows as $ \exp \{c\lambda \}$ as $ \lambda \to \infty $. Thus, we observe the surprising result that the stochastic wave equation is, quite typically, significantly less noise-excitable than its parabolic counterparts.


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Additional Information

Davar Khoshnevisan
Affiliation: Department of Mathematics, University of Utah, Salt Lake City, Utah 84112-0090
Email: davar@math.utah.edu

Kunwoo Kim
Affiliation: Department of Mathematics, University of Utah, Salt Lake City, Utah 84112-0090
Email: kkim@math.utah.edu

DOI: https://doi.org/10.1090/S0002-9939-2015-12517-8
Keywords: Stochastic heat equation, stochastic wave equation, intermittency, non-linear noise excitation
Received by editor(s): September 1, 2013
Received by editor(s) in revised form: March 28, 2014, and April 5, 2014
Published electronically: February 25, 2015
Additional Notes: Research was supported in part by the NSF grant DMS-1006903
Communicated by: Mark M. Meerschaert
Article copyright: © Copyright 2015 American Mathematical Society