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We study the unique existence of weak solutions for initial boundary valueproblems associated with different class of fractional diffusion equationsincluding variable order, distributed order and multiterm fractional diffusionequations. So far, different definitions of weak solutions have been consideredfor these class of problems. This includes definition of solutions in avariational sense and definition of solutions from properties of their Laplacetransform in time. The goal of the present article is to unify these twoapproaches by showing the equivalence of these two definitions. Such propertyallows also to show that the weak solutions under consideration combine theadvantage of these two class of solutions which include representation ofsolutions by a Duhamel type of formula, suitable properties of Laplacetransform of solutions, resolution of the equation in the sense ofdistributions and explicit link with the initial condition.

Equivalence of definitions of solutions for some class of fractional diffusion equations