Initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation

In this paper, the initial-boundary-value problems for the generalized multi-term time-fractional diffusion equation over an open bounded domain G × ( 0 , T ) , G ∈ R n are considered. Based on an appropriate maximum principle that is formulated and proved in the paper, too, some a priory estimates for the solution and then its uniqueness are established. To show the existence of the solution, first a formal solution is constructed using the Fourier method of the separation of the variables. The time-dependent components of the solution are given in terms of the multinomial Mittag-Leffler function. Under certain conditions, the formal solution is shown to be a generalized solution of the initial-boundary-value problem for the generalized time-fractional multi-term diffusion equation that turns out to be a classical solution under some additional conditions. Another important consequence from the maximum principle is a continuously dependence of the solution on the problem data (initial and boundary conditions and a source function) that – together with the uniqueness and existence results – makes the problem under consideration to a well-posed problem in the Hadamard sense.