On Time-Fractional Diffusion Equations with Space-Dependent Variable Order

Abstract

We examine the short and long-time behaviors of time-fractional diffusion equations with variable space-dependent order. More precisely, we describe the time-evolution of the solution to these equations as the time parameter goes either to zero or to infinity.Mathematics subject classification 2010: 35R11, 35B35, 35B38.

Initial-boundary value problem for distributed order time-fractional diffusion equations

“…This statement is of great interest in the analysis of inverse coefficient problems associated with timefractional diffusion equations, see e.g. [10,11,17]. Theorem 1.4.…”

“…As already mentioned in the introduction, the usual definition given in [5] of a weak solution to CO time-fractional diffusion equations, is not suitable for DO time-fractional diffusion equations. Hence we rather follow the strategy implemented in [11] (which is by means of the Laplace transform of tempered distributions), that is recalled below. Let S ′ (R, L 2 (Ω)) := B(S(R, L 2 (Ω)), R) be the space dual to S(R; L 2 (Ω)), put R + := [0, +∞), and denote by S ′ (R + , L 2 (Ω)) := {v ∈ S ′ (R, L 2 (Ω)); supp v ⊂ R + × Ω} the set of distributions in S ′ (R, L 2 (Ω)) that are supported in R + × Ω.…”

“…This being said, we stress out once more that the approach of [9] applies to non-autonomous systems, which is not the case of the analysis presented in this text. Finally, we point out that Definition 1.1 of a weak solution to (1) (as the original function of the solution to the Laplace transform with respect to the time variable of this system) is inspired by the analysis carried out in [11], which is concerned with space dependent variable order (VO) time-fractional diffusion equations. It is well known that the weak solution to CO time-fractional diffusion equations can be expressed in terms of Mittag-Leffler functions, see e.g.…”

Reconstruction and stable recovery of source terms and coefficients appearing in diffusion equations

“…For diffusion equations corresponding to the case α = 1 with time independent source terms, several authors investigated the conditional stability (e.g. [5,37,38] [15,16,17,19,23,31] where some inverse coefficient problems and some related results have been considered.…”

Semi-Markov Models and Motion in Heterogeneous Media

“…The transition probability p(x, y, t) is the fundamental solution to the partial differential equation (5.9), and thus it is interesting to consider some well-posedness issues. The most recent result in this direction can be found in [17] where the authors consider the following Cauchy problem…”