Nonlinear stochastic time-fractional diffusion equations on $\mathbb {R}$: Moments, Hölder regularity and intermittency

Abstract

We study the nonlinear stochastic time-fractional diffusion equations in the spatial domain $\mathbb {R}$, driven by multiplicative space-time 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 slow diffusion equations and fast diffusion equations, respectively. Existence and uniqueness of random field solutions with measure-valued 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 two-parameter Mainardi functions, which are generalizations of the one-parameter Mainardi functions.