Abstract
In this paper, we study the regularity of mild solution for the following fractional abstract Cauchy problem Dt αu(t)=Au(t)+f(t), t ∈ (0,T] u(0)= x0 on a Banach space X with order α ∈ (0,1), where the fractional derivative is understood in the sense of Caputo fractional derivatives. We show that if A generates an analytic α-times resolvent family on X and f ∈ Lp ([0,T];X) for some p > 1/α, then the mild solution to the above equation is in Cα-1/p[o,T] for every o > 0. Moreover, if f is Holder continuous, then so are the Dt αu(t) and Au(t).
Highlights
IntroductionThere are increasing interests on fractional differential equations due to their wide applications in viscoelasticity, dynamics of particles, economic and science et al For more details we refer to [1] [2]
There are increasing interests on fractional differential equations due to their wide applications in viscoelasticity, dynamics of particles, economic and science et al For more details we refer to [1] [2].Many evolution equations can be rewritten as an abstract Cauchy problem, and they can be studied in an unified way
A heat equation with different initial data or boundary conditions can be written as a first order Cauchy problem, in which the governing operator generates a C0-semigroup, and the solution is given by the operation of this semigroup on the initial data
Summary
There are increasing interests on fractional differential equations due to their wide applications in viscoelasticity, dynamics of particles, economic and science et al For more details we refer to [1] [2]. A heat equation with different initial data or boundary conditions can be written as a first order Cauchy problem, in which the governing operator generates a C0-semigroup, and the solution is given by the operation of this semigroup on the initial data. In this paper we are mainly interested in the Hölder regularity for abstract Cauchy problems of fractional order. Pazy [4] considered the regularity for the abstract Cauchy problem of first order: u′(t) =Au(t) + f (t), t ∈[0,T ]. Dt gave similar results for fractional differential equations with order α ∈ (1, 2). In this paper we will extend their results to fractional Cauchy problems with order in (0,1).
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