NONLINEAR STOCHASTIC TIME-FRACTIONAL DIFFUSION EQUATIONS ON R: MOMENTS, HÖLDER REGULARITY AND INTERMITTENCY

We study the nonlinear stochastic time-fractional diffusion equations in the spatial domain R, driven by multiplicative space-time white noise. The fractional index β varies continuously from 0 to 2. The case β = 1 (resp. β = 2) corresponds to the stochastic heat (resp. wave) equation. The cases β ∈ ]0, 1[ and β ∈ ]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 ≥ 2) are obtained, which are expressed using a kernel function 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.