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

We study the unique existence of weak solutions for initial boundary value problems associated with different class of fractional diffusion equations including variable order, distributed order, and multiterm fractional diffusion equations. So far, different definitions of weak solutions have been considered for these class of problems. This includes definition of solutions in a variational sense and definition of solutions from properties of their Laplace transform in time. The goal of this article is to unify these two approaches by showing the equivalence of these two definitions. Such a property allows also to show that the weak solutions under consideration combine the advantages of these two classes of solutions, which include representation of solutions by a Duhamel type of formula, suitable properties of Laplace transform of solutions, resolution of the equation in the sense of distributions, and explicit link with the initial condition.