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In this paper, the initial-boundary-value problems for the generalized multi-term timefractional 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 initialboundary-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) thattogether with the uniqueness and existence results -makes the problem under consideration to a well-posed problem in the Hadamard sense.

In the paper, a maximum principle for the generalized time-fractional diffusion equation over an open bounded domain G × (0, T ), G ⊂ R n is formulated and proved. The proof of the maximum principle is based on an extremum principle for the Caputo-Dzherbashyan fractional derivative that is given in the paper, too. The maximum principle is then applied to show that the initial-boundary-value problem for the generalized time-fractional diffusion equation possesses at most one classical solution and this solution continuously depends on the initial and boundary conditions.

a b s t r a c tIn this paper, some uniqueness and existence results for the solutions of the initialboundary-value problems for the generalized time-fractional diffusion equation over an open bounded domain G × (0, T ), G ⊂ R n are given. To establish the uniqueness of the solution, a maximum principle for the generalized time-fractional diffusion equation is used. In turn, the maximum principle is based on an extremum principle for the CaputoDzherbashyan fractional derivative that is considered in the paper, too. Another important consequence of the maximum principle is the continuous dependence of the solution on the problem data. To show the existence of the solution, the Fourier method of the variable separation is used to construct a formal solution. Under certain conditions, the formal solution is shown to be a generalized solution of the initial-boundary-value problem for the generalized time-fractional diffusion equation that turns out to be a classical solution under some additional conditions.