Dynamics of Partial Differential Equations

Volume 13 (2016)

Number 2

Fractional abstract Cauchy problem with order $\alpha \in (1,2)$

Pages: 155 – 177

DOI: https://dx.doi.org/10.4310/DPDE.2016.v13.n2.a4


Ya-Ning Li (College of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, China; and School of Mathematics and Statistics, Lanzhou University, Lanzhou, China)

Hong-Rui Sun (School of Mathematics and Statistics, Lanzhou University, Lanzhou, China)

Zhaosheng Feng (Department of Mathematics, University of Texas–Rio Grande Valley, Edinburg, Tx., U.S.A.)


In this paper, we deal with a class of fractional abstract Cauchy problems of order $\alpha \in (1,2)$ by introducing an operator $S_{\alpha}$ which is defined in terms of the Mittag-Leffler function and the curve integral. Some nice properties of the operator $S_{\alpha}$ are presented. Based on these properties, the existence and uniqueness of mild solution and classical solution to the inhomogeneous linear and semilinear fractional abstract Cauchy problems is established accordingly. The regularity of mild solution of the semilinear fractional Cauchy problem is also discussed.


fractional Cauchy problem, Mittag-Leffler function, fractional resolvent, mild solution, sectorial operator

2010 Mathematics Subject Classification

Primary 26A33, 35K90. Secondary 47A10, 47D06.

Published 23 June 2016