The existence of mild and classical solutions for time fractional Fokker-Planck equations

Time fractional Fokker-Planck equations can be used to describe the subdiffusion in an external time-and space-dependent force field F(t, x). In this paper, we convert it to the form of the following problems partial derivative(u)(t) - kappa(alpha)partial derivative(1-alpha)(t) Delta u = del . (F partial derivative(1-alpha)(t)u) + f, where alpha is an element of (0, 1). We obtain some results on existence and uniqueness of mild solutions allowing the "working space" that may have low regularity. Secondly, we analyze the relationship between "working space" and the value range of a when investigating the problem of classical solutions. Finally, by constructing a suitable weighted Holder continuous function space, the existence of classical solutions is derived without the restriction on alpha is an element of (1/2, 1).