Intermittence and nonlinear parabolic stochastic partial differential equations

We consider nonlinear parabolic SPDEs of the form. partial derivative(t)u = Lu + sigma( u)(w) over dot , where (w) over dot denotes space-time white noise, sigma : R -> R is [globally] Lipschitz continuous, and L is the L-2-generator of a Levy process. We present precise criteria for existence as well as uniqueness of solutions. More significantly, we prove that these solutions grow in time with at most a precise exponential rate. We establish also that when s is globally Lipschitz and asymptotically sublinear, the solution to the nonlinear heat equation is "weakly intermittent," provided that the symmetrization of L is recurrent and the initial data is sufficiently large. Among other things, our results lead to general formulas for the upper second-moment Liapounov exponent of the parabolic Anderson model for L in dimension (1 + 1). When L = kappa partial derivative(xx) for kappa > 0, these formulas agree with the earlier results of statistical physics [28; 32; 33], and also probability theory [1; 5] in the two exactly-solvable cases. That is when u(0) = delta(0) or u(0) equivalent to 1; in those cases the moments of the solution to the SPDE can be computed [1].