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Hölder regularity for non-autonomous fractional evolution equations

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Abstract

This paper concerns a Hölder regularity result for non-autonomous fractional evolution equations (NFEEs) of the form \(^C\!D^\alpha _t u(t)+A(t)u(t)=f(t)\) in the sense of Caputo’s fractional derivative. Under the assumption of the Acquistapace-Terremi conditions, we get a representation of solution that is closer to standard integral equation. A pair of families of solution operators will be constructed in terms of the Mittag-Leffler function, the Mainardi Wright-type function, the analytic semigroups generated by closed dense operator \(A(\cdot )\) and two variation of parameters formulas. By a suitable definition of classical solutions, the solvability of the Cauchy problem is established. Moreover, we also prove the Hölder regularity of classical solutions.

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Acknowledgements

The work was supported by National Natural Science Foundation of China (12101142, 12071396).

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Authors and Affiliations

  1. College of Mathematics and Information Science, Guangxi University, Nanning, 530004, China

    Jia Wei He

  2. Faculty of Mathematics and Computational Science, Xiangtan University, Hunan, 411105, China

    Yong Zhou

  3. Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, Faculty of Science, King Abdulaziz University, Jeddah, 21589, Saudi Arabia

    Yong Zhou

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  1. Jia Wei He
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  2. Yong Zhou
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Correspondence to Yong Zhou.

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He, J.W., Zhou, Y. Hölder regularity for non-autonomous fractional evolution equations. Fract Calc Appl Anal 25, 378–407 (2022). https://doi.org/10.1007/s13540-022-00019-1

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