A Survey on the Recent Results Regarding Maximum Principles for the Time-Fractional Diffusion Equations

In this chapter, a survey on the recent results regarding the maximum principles for the time-fractional diffusion equations is presented. We formulate the maximum principles for the time-fractional partial differential equations with both the Caputo fractional derivative and the Riemann-Liouville fractional derivative. Along with the single-term time-fractional differential equations, the multi-term equations and the equations of the distributed order are considered. We also discuss some important applications of the maximum principles for the time-fractional diffusion equations including a priori estimates for solutions of the initial-boundaryvalue problems for these equations and uniqueness of their solutions.

Keywords: Fractional derivatives, Time-fractional diffusion equation, Initialboundary- value problems, Maximum principle, A priori estimates, Uniqueness of solutions.