Well-Posedness for Weak and Strong Solutions of Non-Homogeneous Initial Boundary Value Problems for Fractional Diffusion Equations

Abstract

We study the well-posedness for initial boundary value problems associated with time fractional diffusion equations with non-homogenous boundary and initial values. We consider both weak and strong solutions for the problems. For weak solutions, we introduce a definition of solutions which allows to prove the existence of solution to the initial boundary value problems with non-zero initial and boundary values and non-homogeneous source terms lying in some negative-order Sobolev spaces. For strong solutions, we introduce an optimal compatibility condition and prove the existence of the solutions. We introduce also some sharp conditions guaranteeing the existence of solutions with more regularity in time and space.

Weak Solutions for Time-Fractional Evolution Equations in Hilbert Spaces

“…We show that the notion of weak solutions we propose may be strengthened by the spectral approach followed in [3,4]. Indeed, we prove that the weak solutions may be written by means of an explicit representation formula through expansions involving the Mittag-Leffler functions and the eigenvalues of the spatial operator.…”

“…Formula (4) gives an explicit representation through expansions involving the Mittag-Leffler functions, the eigenvalues λ n and the eigenfunctions e n of the spatial operator. This representation formula is suggested by the spectral approach given in [3,4], where the authors deal with uniformly elliptic operators in a different setting. It is worthwhile to mention the paper [5] (see also references therein), where the authors give the representation of classical solutions by means of the α-resolvent family.…”

Recovering the potential and order in one-dimensional time-fractional diffusion with unknown initial condition and source *

“…Now we study the direct problem (1.1), especially the solution representation. One distinct feature of problem (1.1) is that it involves a nonzero Neumann boundary condition, which has not been extensively studied in the literature ( [25,37] for relevant works). Following [37], we exploit the one-dimensional nature of problem (1.1), and derive a series representation of the solution u.…”

Fractional derivatives and time-fractional ordinary differential equations in $L^p$-space

“…There are several studies and here we refer to In Gorenflo, Luchko and Yamamoto [6] and Kubica, Ryszewska and Yamamoto [15], Yamamoto [23], where one constructs the fractional derivative operator in Sobolev spaces based on L 2 (0, T ) and apply it to fractional differential equations including partial differential equations in spatial variables. Also see Kian [11], Kian and Yamamoto [12], Luchko and Yamamoto [18], Sakamoto and Yamamoto [21], Zacher [24]. The method in those works relies on the structure of Hilbert spaces, that is, p = 2.…”