Analysis of fractional diffusion equations of distributed order: Maximum principles and their applications

In this paper, we formulate and prove the weak and strong maximum principles for a general parabolic-type fractional differential operator with the Riemann-Liouville time-fractional derivative of distributed order. The proofs of the maximum principles are based on an estimate of the Riemann-Liouville fractional derivative at its maximum point that was recently derived by the authors. Some a priori norm estimates for solutions to initial-boundary value problems for linear and nonlinear fractional diffusion equations of distributed order and uniqueness results for these problems are presented.