  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 the present article is to unify these two
approaches by showing the equivalence of these two definitions. Such property
allows also to show that the weak solutions under consideration combine the
advantage of these two class 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.
