Non-linear noise excitation and intermittency under high disorder

**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 KimDepartment 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