H\"older regularity of the nonlinear stochastic time-fractional slow and fast diffusion equations on $\mathbb{R}^d$

TL;DR

This paper proves the H"older continuity of solutions to nonlinear time-fractional diffusion equations driven by space-time white noise, extending previous results by removing a key restrictive condition.

Contribution

It establishes H"older regularity for a broader class of nonlinear stochastic fractional diffusion equations without the previous constraint on parameters.

Findings

H"older continuity of solutions under new parameter conditions

Extension of regularity results to all b2(0,2)

Removal of previous restrictions on b2 and b3

Abstract

In this paper, we use local fraction derivative to show the H\"older continuity of the solution to the following nonlinear time-fractional slow and fast diffusion equation: $$\left(\partial^\beta+\frac{\nu}{2}(-\Delta)^{\alpha/2}\right)u(t,x) = I_t^\gamma\left[\sigma\left(u(t,x)\right)\dot{W}(t,x)\right],\quad t>0,\: x\in\mathbb{R}^d,$$ where $\dot{W}$ is the space-time white noise, $\alpha\in(0,2]$, $\beta\in(0,2)$, $\gamma\ge 0$ and $\nu>0$, under the condition that $2(\beta+\gamma)-1-d\beta/\alpha>0$. The case when $\beta+\gamma\le 1$ has been obtained in \cite{ChHuNu19}. In this paper, we have removed this extra condition, which in particular includes all cases for $\beta\in(0,2)$.