## Non-linear noise excitation and intermittency under high disorder

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- by Davar Khoshnevisan and Kunwoo Kim PDF
- Proc. Amer. Math. Soc.
**143**(2015), 4073-4083 Request permission

## 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 $\inf _z|\sigma (z)/z|>0$], 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.## References

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## Additional Information

**Davar Khoshnevisan**- Affiliation: Department of Mathematics, University of Utah, Salt Lake City, Utah 84112-0090
- MR Author ID: 302544
- 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
- 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
- © Copyright 2015 American Mathematical Society
- 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
- MathSciNet review: 3359595