Intermittency properties of space-time fractional stochastic partial differential equations

Abstract

This dissertation focuses on the analyses of the non-linear time-fractional stochastic reaction-diffusion equations of the type \begin{equation}\label{abstract-eq} \partial^\beta_tu_t(x)=-\nu(-\Delta)^{\alpha/2} u_t(x)+I^{1-\beta}[b(u)+ \sigma(u)\stackrel{\cdot}{F}(t,x)] \end{equation} in $(d+1)$ dimensions, where $\nu>0, \beta\in (0,1)$, $\alpha\in (0,2]$ and $d$ is a positive integer. The operator $\partial^\beta_t$ is the Caputo fractional derivative while $-(-\Delta)^{\alpha/2} $ is the generator of an isotropic $\alpha$-stable L\'evy process and $I^{1-\beta}$ is the Riesz fractional integral operator. The forcing noise denoted by $\stackrel{\cdot}{F}(t,x)$ is a Gaussian or white noise. These equations might be used as a model for materials with random thermal memory.

The first part of the dissertation studies {\it intermittency fronts} for the solution of the stochastic equation of Eq.\eqref{abstract-eq} when $b\equiv0$. Under some appropriate conditions on the parameters we prove that solutions to the initial value problem of Eq.\eqref{abstract-eq} with nonempty measurable initial function with compact support and strictly positive on an open subset of $(0,\infty)^d $ have positive intermittency lower front. Furthermore, we also identified the parameters regions ensuring that the solutions to the initial value problem of Eq.\eqref{abstract-eq} with the same condition on the initial function also have finite intermittency upper front. Our results recovers as particular cases some known results in the literature. For example, Mijena and Nane proved in \cite{JebesaAndNane1} that : (i) absolute moments of the solutions of this equation grow exponentially; and (ii) the distances to the origin of the farthest high peaks of those moments grow exactly linearly with time. The last result was proved under the assumptions $\alpha=2$ and $d=1.$ Here, we extend this result to the case $\alpha=2$ and $d\in\{1,2,3\}.$

Next, we study the phenomena of finite-time blow up and non-existence of solutions of \eqref{abstract-eq}. In particular, when the term $\sigma(u)$ satisfies $\sigma(u)\geq |x|^{1+\gamma}$ for some positive number $\gamma$, we prove that solution to the initial value problem of Eq.\eqref{abstract-eq} with strictly positive initial distribution have infinite second moment for $t$ large enough. We derive non-existence (blow-up) of global random field solutions under some additional conditions, most notably on $b$, $\sigma$ and the initial condition. Our results complement those of P. Chow in \cite{chow2}, \cite{chow1}, and Foondun et al. in \cite{Foondun-liu-nane}, \cite{foondun-parshad} among others.