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

Abstract

In this paper, we use local fraction derivative to show the Hölder continuity of the solution to the following nonlinear time-fractional slow and fast diffusion equation: $$\left({\partial }^{\beta }+\frac{\nu }{2}(-\Delta {)}^{\alpha /2}\right)u(t,x)=I_t^{\gamma }\left[\sigma \left(u(t,x)\right)\dot{W}(t,x)\right],t>0,x\in {\mathbb{R}}^d,$$ 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 $\beta \in (0,2)$.

17 pages, 2 figures