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

Abstract

In this paper, we study the space-time Hölder continuity of the solution to the following nonlinear time-fractional slow and fast diffusion equation: $$\begin{aligned} \left( \partial ^\beta +\frac{\nu }{2}(-{\varDelta })^{\alpha /2}\right) u(t,x) = I_t^\gamma \left[ \sigma \left( u(t,x)\right) {\dot{W}}(t,x)\right] ,\quad t>0,\ x\in {\mathbb {R}}^d, \end{aligned}$$where \({\dot{W}}\) is the space-time white noise, \(\alpha \in (0,2]\), \(\beta \in (0,2)\), \(\gamma \ge 0\) and \(\nu >0\). The existence/uniqueness of a random field solution has been obtained in [9] under the condition that \(2(\beta +\gamma )-1-d\beta /\alpha >0\). The Hölder regularity of the solution has been obtained in the same reference, but only for the case \(\beta +\gamma \le 1\). In this paper, we use the idea from the local fractional derivative to establish the Hölder regularity of the solution for all possible cases – \(\beta \in (0,2)\), which in particular recovers the special case in [9] when \(\beta \in (0,1-\gamma ]\). As a rather surprising consequence, when \(\gamma =0\), \(\alpha =2\) and \(\beta \) is close to 2, the space and time Hölder exponents are both to \(1-\), which is different from the known Hölder exponents for the stochastic wave equation which are \((1/2)-\).