JAMP  Vol.6 No.1 , January 2018
Hölder Regularity for Abstract Fractional Cauchy Problems with Order in (0,1)
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).
Cite this paper: Li, C. and Li, M. (2018) Hölder Regularity for Abstract Fractional Cauchy Problems with Order in (0,1). Journal of Applied Mathematics and Physics, 6, 310-319. doi: 10.4236/jamp.2018.61030.

[1]   Hilfer, R. (2000) Applications of Fractional Calculus in Physics. World Scientific, Singapore.

[2]   Tarasov, V. (2011) Fractional Dynamics: Applications of Fractional Calculus to Dynamics of Particles, Fields and Media. Springer, New York.

[3]   Arendt, W., Batty, C.J.K., Hieber, M. and Neubrander, F. (2010) Vector-Valued Laplace Transforms and Cauchy Problems. Birkhäuser,.

[4]   Pazy, A. (1983) Semigroups of Linear Operators and Applications to Partial Differential Equations. Springer, New York.

[5]   Prüss, J. (1993) Evolutionary Integral Equations and Applications. Birkhäuser, Basel.

[6]   Bajlekova, E.G. (2001) Fractional Evolution Equations in Banach Spaces. Ph.D. Thesis, Department of Mathematics, Eindhoven University of Technology.

[7]   EI-Boral, M.M. (2004) Semi-group and Some Nonlinear Fractional Differential Equations. Appl. Math. Comput., 149, 823-831.

[8]   EI-Boral, M.M. (2002) Some Probability Densities and Fundamental Solutions of Fractional Evolution Equations. Chaos Solitons Fractals, 14, 433-440.

[9]   EI-Boral, M.M., EI-Nadi, K.E. and EI-Akabawy, E.G. (2010) On Some Fractional Evolution Equations. Comput. Math. Anal., 59, 1352-1355.

[10]   Li, M., Chen, C. and Li, F.B. (2010) On Fractional Powers of Generators of Fractional Resolvent Families. J. Funct. Anal., 259, 2702-2726.

[11]   Akrami, M.H. and Erjaee, G.H. (2014) Existence of Positive Solutions for a Class of Fractional Differential Equation Systems with Multi-Point Boundary Conditions. Georgian Math. J., 21, 379-386.

[12]   Li, F.B. and Li, M. (2013) On Maximal Regularity and Semivariation of Times Resolvent Families. Pure Math., 3, 680-684.

[13]   Clément, P., Gripenberg, G. and Londen, S-O. (2000) Schauder Estimates for Equations with Fractional Derivatives. Trans. Amer. Math. Soc., 352, 2239-2260.

[14]   Clément, P., Gripenberg, G. and Londen, S-O. (1998) Hölder Regularity for a Linear Fractional Evolution Equations, Topics in Nonlinear Analysis. The Herbert Amann Anniversary Volume, Birkhäuser, Basel.

[15]   Fernandez-Real, X. and Ros-Oton, X. (2016) Boundary Regularity for the Fractional Heat Equation, RACSAM110.

[16]   Li, Y. (2015) Regularity of Mild Solutions for Fractional Abstract Cauchy Problem with Order , Z. Angew. Math. Phys., 66, 3283-3298.

[17]   Samko, S.G., Klibas, A.A. and Marichev, O.I. (1992) Fractional Integrals and Derivatives, Theory and Applications. Gordon and Breach Science Publishers.

[18]   Crandall, M.G. and Pazy, A. (1968/1969) On the Differentiability of Weak Solutions of a Differential Euqation in Banach Space. J. Math. Mech., 18, 1007-1016.