Nonlinear stochastic timefractional diffusion equations on $\mathbb{R}$: moments, Hölder regularity and intermittency
Abstract
We study the nonlinear stochastic timefractional diffusion equations in the spatial domain $\mathbb{R}$, driven by multiplicative spacetime white noise. The fractional index $\beta$ varies continuously from $0$ to $2$. The case $\beta=1$ (resp. $\beta=2$) corresponds to the stochastic heat (resp. wave) equation. The cases $\beta\in \:]0,1[\:$ and $\beta\in \:]1,2[\:$ are called {\it slow diffusion equations} and {\it fast diffusion equations}, respectively. Existence and uniqueness of random field solutions with measurevalued initial data, such as the Dirac delta measure, are established. Upper bounds on all $p$th moments $(p\ge 2)$ are obtained, which are expressed using a kernel function $\mathcal{K}(t,x)$. The second moment is sharp. We obtain the Hölder continuity of the solution for the slow diffusion equations when the initial data is a bounded function. We prove the weak intermittency for both slow and fast diffusion equations. In this study, we introduce a special function, the {\it twoparameter Mainardi functions}, which are generalizations of the oneparameter Mainardi functions.
 Publication:

arXiv eprints
 Pub Date:
 October 2014
 arXiv:
 arXiv:1410.1911
 Bibcode:
 2014arXiv1410.1911C
 Keywords:

 Mathematics  Probability;
 60H15
 EPrint:
 42 pages, 8 figures