NON-LINEAR NOISE EXCITATION AND INTERMITTENCY UNDER HIGH DISORDER

Abstract

Consider the semilinear heat equation partial derivative(t)u = partial derivative(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) vertical bar sigma(z)/z vertical bar > 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 -> infinity. 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 R, the energy of the solution grows as exp{c lambda} as lambda -> infinity. Thus, we observe the surprising result that the stochastic wave equation is, quite typically, significantly less noise-excitable than its parabolic counterparts.