Intermittence and nonlinear parabolic stochastic partial differential equations

2008/05/05 by Foondun, Mohammud, Khoshnevisan, Davar · 4 citations
#60H15 #82B44 #FOS: Mathematics #Probability (math.PR)

paper · doi:10.48550/arxiv.0805.0557

Abstract

We consider nonlinear parabolic SPDEs of the form ∂_t u=\sL u + σ(u) w, where w denotes space-time white noise, σ:\R→\R is [globally] Lipschitz continuous, and \sL is the L²-generator of a Lévy 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 σ is globally Lipschitz and asymptotically sublinear, the solution to the nonlinear heat equation is ``weakly intermittent,'' provided that the symmetrization of \sL 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 \sL in dimension (1+1). When \sL=κ∂_xx for κ>0, these formulas agree with the earlier results of statistical physics \citeKardar,KrugSpohn,LL63, and also probability theory \citeBC,CM94 in the two exactly-solvable cases where u₀=δ₀ and u₀≡ 1.

Cited by

• On the local well-posedness of randomly forced reaction-diffusion equations with L² initial data and a superlinear reaction term
• Lyapunov exponents and growth indices for fractional stochastic heat equations with space-time Lévy white noise
• A stochastic heat equation with non-locally Lipschitz coefficients
• Sharp moment and upper tail asymptotics for the critical 2d Stochastic Heat Flow

Related

• Remarks on non-linear noise excitability of some stochastic heat equations
• On non-existence of global solutions to a class of stochastic heat\n equations
• Intermittency for the Hyperbolic Anderson Model with rough noise in space
• Dissipation in Parabolic SPDEs II: Oscillation and decay of the solution
• Points of slow growth for parabolic SPDEs