  The diffusion system with time-fractional order derivative is of great
importance mathematically due to the nonlocal property of the fractional order
derivative, which can be applied to model the physical phenomena with memory
effects. We consider an initial-boundary value problem for the time-fractional
diffusion equation with inhomogenous Robin boundary condition. Firstly, we show
the unique existence of the weak/strong solution based on the eigenfunction
expansions, which ensures the well-posedness of the direct problem. Then, we
establish the Hopf lemma for time-fractional diffusion operator, generalizing
the counterpart for the classical parabolic equation. Based on this new Hopf
lemma, the maximum principles for this time-fractional diffusion are finally
proven, which play essential roles for further studying the uniqueness of the
inverse problems corresponding to this system.
