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

## Davar Khoshnevisan and Kunwoo Kim

Abstract. Consider the semilinear heat equation $$\partial_t u = \partial^2_x u + \lambda\sigma(u)\xi$$ on the interval $$[0\,,1]$$ 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 $$\inf_{z\in{\bf R}}|\sigma(z)/z|>0$$], 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 $$\lambda\to\infty$$. When the initial function is bounded uniformly away from zero and the Dirichlet boundary condition is replaced by a Neumann condition, then the $$L^2$$-energy of the solution is in fact shown to be of sharp exponential order $$\exp\{c\lambda^4\}$$. We show also that, for a large family of one-dimensional randomly-forced wave equations, 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.

Keywords. The stochastic heat equation; the stochastic wave equation; intermittency; non-linear noise excitation.

AMS Classification (2000). Primary 60H15, 60H25; Secondary 35R60, 60K37, 60J30, 60B15.

Support. Research supported in part by a grant from the U.S. National Science Foundation.

Pre/E-Prints. This paper is available in

• Pdf Format (New Version; contains an important correction: March 4, 2013)

 Davar Khoshnevisan & Kunwoo Kim Department of Mathematics University of Utah 155 S, 1400 E JWB 233 Salt Lake City, UT 84112-0090, U.S.A. davar@math.utah.edu & kkim@math.utah.edu

Last Update: March 4, 2013
© 2013 - Davar Khoshnevisan & Kunwoo Kim