Nonlinear stochastic time-fractional diffusion equations on $\mathbb{R}$: moments, H\"older regularity and intermittency

TL;DR

This paper investigates nonlinear stochastic time-fractional diffusion equations on the real line, analyzing moments, regularity, and intermittency for various fractional indices, and introduces new special functions called Mainardi functions.

Contribution

It establishes existence, uniqueness, moment bounds, regularity, and intermittency results for these equations, and introduces the two-parameter Mainardi functions as a novel analytical tool.

Findings

Existence and uniqueness of solutions with measure-valued initial data.

Sharp second moment bounds expressed via kernel functions.

Proven weak intermittency for slow and fast diffusion cases.

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 {\it slow diffusion equations} and {\it 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 H\"older 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 two-parameter Mainardi functions}, which are generalizations of the one-parameter Mainardi functions.