Intermittence and nonlinear parabolic stochastic partial differential equations
Mohammud Foondun, University of Utah
Davar Khoshnevisan, University of Utah
Abstract
We consider nonlinear parabolic
SPDEs of the form
$partial_t u=sL u + sigma(u)dot w$, where
$dot w$ denotes space-time white noise,
$sigma:RtoR$ is [globally] Lipschitz continuous,
and $sL$ is the $L^2$-generator of a L'evy 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 $sigma$ 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
$sL=kappapartial_{xx}$ for $kappa>0$, these formulas
agree with the earlier results of statistical physics
cite{Kardar,KrugSpohn,LL63}, and also probability theory
cite{BC,CM94} in the two exactly-solvable cases. That is
when $u_0=delta_0$ or $u_0equiv 1$; in those cases
the moments of the solution to the SPDE can be computed
cite{BC}.
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