Cauchy problem for fractional diffusion equations

Abstract

We consider an evolution equation with the regularized fractional derivative of an order aAð0; 1Þ with respect to the time variable, and a uniformly elliptic operator with variable coefficients acting in the spatial variables. Such equations describe diffusion on inhomogeneous fractals. A fundamental solution of the Cauchy problem is constructed and investigated. r

Existence of Asymptotically Almost Automorphic Mild Solutions of Semilinear Fractional Differential Equations

“…Indeed, due to their applications in fields of science where characteristics of anomalous diffusion are presented, type (6) equations are attracting increasing interest (cf. [68][69][70] and references therein). For example, anomalous diffusion in fractals [69] or in macroeconomics [71] has been recently well studied in the setting of fractional Cauchy problems like (6).…”

“…[68][69][70] and references therein). For example, anomalous diffusion in fractals [69] or in macroeconomics [71] has been recently well studied in the setting of fractional Cauchy problems like (6). For this reason, (6) has gotten a considerable attention in recent years (cf.…”

“…For this reason, (6) has gotten a considerable attention in recent years (cf. [51][52][53][54][55][56][57][58][59][60][61][62][63][64][68][69][70][71] and the references therein).…”

“…Set Θ fl { ( ) ∈ 0 (R, ) : |‖ ( )‖| ≤ }. For the above V( ), define Γ fl Γ 1 + Γ 2 on 0 (R, ) as (69) and prove that Γ maps Θ 0 into itself, where 0 is a given constant. Firstly, from (97) it follows that, for all ∈ R, ( ) ∈ ,…”

Existence and uniqueness of the solution for a time-fractional diffusion equation

“…In literature mainly the Cauchy problems for these equations has been considered until now (see [9,1,12] and references therein).…”