Maximum principle for the time-space fractional diffusion equations

Recently, with the development of fractional differential equations (FDEs), the maximum principles for FDEs are still an open problem. In this paper, we will consider a family of the space-time Riemmann-Liouville fractional differential equations over an open bounded domain. In order to derive the maximum principles for space-time Riemmann-Liouville fractional differential equations, the Riemann-Liouville fractional derivatives at extreme points was discussed and established. Then, the maximum principles were proved by applying those results. Finally, maximum principles are employed to show that the initial-boundary-value problem for the possesses at most one solution and this solution continuously depends on the initial condition. © 2015 American Scientific Publishers.