TITLE:
Hölder Regularity for Abstract Fractional Cauchy Problems with Order in (0,1)
AUTHORS:
Chenyu Li, Miao Li
KEYWORDS:
Fractional Cauchy Problem, Fractional Resolvent Family, Generator, Regularity, Hölder Continuity
JOURNAL NAME:
Journal of Applied Mathematics and Physics,
Vol.6 No.1,
January
31,
2018
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[ò,T] for every ò > 0. Moreover, if f is Hölder continuous, then so are the Dt αu(t) and Au(t).