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

Abstract

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.

Solving time-fractional diffusion equations with a singular source term

“…, generalizes the ones given in [4,13,18,21,22] and in the references therein, in the context of more specific diffusion processes. Moreover, the representation formula of the weak solution used in [13,18,22] is a byproduct of the Duhamel formula (2.3) that is established in theorem 2.1 for a wider set of source terms lying in S ′ + (R, L 2 (Ω)). It is worth pointing out that theorem 2.1 holds provided that the elliptic part A of the diffusion equation (1.1) is symmetric.…”

“…[4,20,30] and the references therein. Similarly, several techniques were used in [18,[22][23][24] to build a solution to variable-order, distributed or multi-term time-fractional processes, and we refer the reader to [13] for a global comparative analysis of these different approaches. All the above mentioned works assume that the source term F is within the class L 1 loc (R + , L 2 (Ω)) but, recently, the well-posedness of constant-order time-fractional diffusion systems was examined in [34] when t → F(t, •) lies in a negative order Sobolev space.…”

Inverse source problem with a posteriori boundary measurement for fractional diffusion equations

“…With reference to Kian and Yamamoto [1,2], one can check that (1.4) admits a unique weak solution lying in L 1 loc (R + ; H s (Ω)), 1 ≤ s < 2. Fixing T ∈ R + , in the present article, we assume that the source term F satisfies the following condition:…”

“…According to Kian and Yamamoto [1,2], assuming that F ∈ L 1 (R + ; L 2 (Ω)), one can check that problem (1.4) admits a unique weak solution in the sense of Definition 2.1.…”