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.
