# American Institute of Mathematical Sciences

August  2020, 13(8): 2259-2270. doi: 10.3934/dcdss.2020187

## Fractional Cauchy problems and applications

 1 Department of Mathematics, The University of Jordan, Amman, Jordan 2 Dipartimento di Matematica, Università di Bologna, Piazza di Porta S. Donato, 5 – 40126 Bologna, Italy 3 Takarazuka, Hirai Sanso 12-13,665-0817, Japan

Received  August 2018 Revised  April 2019 Published  August 2020 Early access  November 2019

We are devoted to fractional abstract Cauchy problems. Required conditions on spaces and operators are given guaranteeing existence and uniqueness of solutions. An inverse problem is also studied. Applications to partial differential equations are given to illustrate the abstract fractional degenerate differential problems.

Citation: Mohammed AL Horani, Mauro Fabrizio, Angelo Favini, Hiroki Tanabe. Fractional Cauchy problems and applications. Discrete and Continuous Dynamical Systems - S, 2020, 13 (8) : 2259-2270. doi: 10.3934/dcdss.2020187
##### References:
 [1] M. Al Horani, M. Fabrizio, A. Favini and H. Tanabe, Direct and inverse problems for degenerate differential equations, Ann. Univ. Ferrara Sez. VII Sci. Mat., 64 (2018), 227-241.  doi: 10.1007/s11565-018-0303-9. [2] E. G. Bazhlekova, Fractional evolution equations in Banach spaces, Dissertation, Eindhoven University of Technology in Eindhoven, 2001. [3] A. Favaron, A. Favini and H. Tanabe, Perturbation methods for inverse problems on degenerate differential equations, preprint. [4] A. Favini, A. Lorenzi, G. Marinoschi and H. Tanabe, Perturbation methods and identification problems for degenerate evolution systems, in Advances in Mathematics, Ed. Acad. Romȃne, Bucharest, 2013,145–156. [5] A. Favini, A. Lorenzi and H. Tanabe, Degenerate integro-differential equations of parabolic type with Robin boundary conditions: $L^p$-theory, J. Math. Anal. Appl., 447 (2017), 579-665.  doi: 10.1016/j.jmaa.2016.10.029. [6] A. Favini, A. Lorenzi and H. Tanabe, Direct and inverse degenerate parabolic differential equations with multi-valued operators, Electron. J. Differential Equations, 2015, 22pp. [7] A. Favini, A. Lorenzi and H. Tanabe, Singular integro-differential equations of parabolic type, Adv. Differential Equations, 7 (2002), 769-798. [8] A. Favini and H. Tanabe, Degenerate differential equations of parabolic type and inverse problems, Proceedings of Seminar on Partial Differential Equations, Osaka University, Osaka, 2015, 89–100. [9] A. Favini and A. Yagi, Multivalued linear operators and degenerate evolution equations, Ann. Mat. Pura Appl., 163 (1993), 353-384.  doi: 10.1007/BF01759029. [10] A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces, Monographs and Textbooks in Pure and Applied Mathematics, 215, Marcel Dekker, Inc., New York, 1999. [11] D. Guidetti, On maximal regularity for the Cauchy-Dirichlet mixed parabolic problem with fractional time derivative, Bruno Pini Mathematical Analysis Seminar, 9, Univ. Bologna, Bologna, 2018,147–157. [12] J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Die Grundlehren der Mathematischen Wissenschaften, 2, Springer-Verlag, New York-Heidelberg, 1972. [13] J. L. Lions and J. Peetre, Sur une classe d'espaces d'interpolation, Inst. Hautes Études Sci. Publ. Math., 19 (1964), 5–68. [14] A. Lorenzi, An Introduction to Identification Problems Via Functional Analysis, Inverse and Ill-Posed Problems Series, Utrecht, Netherlands, 2001. doi: 10.1017/CBO9781139168724. [15] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland Mathematical Library, 18, North-Holland Publishing Co., Amsterdam-New York, 1978.

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##### References:
 [1] M. Al Horani, M. Fabrizio, A. Favini and H. Tanabe, Direct and inverse problems for degenerate differential equations, Ann. Univ. Ferrara Sez. VII Sci. Mat., 64 (2018), 227-241.  doi: 10.1007/s11565-018-0303-9. [2] E. G. Bazhlekova, Fractional evolution equations in Banach spaces, Dissertation, Eindhoven University of Technology in Eindhoven, 2001. [3] A. Favaron, A. Favini and H. Tanabe, Perturbation methods for inverse problems on degenerate differential equations, preprint. [4] A. Favini, A. Lorenzi, G. Marinoschi and H. Tanabe, Perturbation methods and identification problems for degenerate evolution systems, in Advances in Mathematics, Ed. Acad. Romȃne, Bucharest, 2013,145–156. [5] A. Favini, A. Lorenzi and H. Tanabe, Degenerate integro-differential equations of parabolic type with Robin boundary conditions: $L^p$-theory, J. Math. Anal. Appl., 447 (2017), 579-665.  doi: 10.1016/j.jmaa.2016.10.029. [6] A. Favini, A. Lorenzi and H. Tanabe, Direct and inverse degenerate parabolic differential equations with multi-valued operators, Electron. J. Differential Equations, 2015, 22pp. [7] A. Favini, A. Lorenzi and H. Tanabe, Singular integro-differential equations of parabolic type, Adv. Differential Equations, 7 (2002), 769-798. [8] A. Favini and H. Tanabe, Degenerate differential equations of parabolic type and inverse problems, Proceedings of Seminar on Partial Differential Equations, Osaka University, Osaka, 2015, 89–100. [9] A. Favini and A. Yagi, Multivalued linear operators and degenerate evolution equations, Ann. Mat. Pura Appl., 163 (1993), 353-384.  doi: 10.1007/BF01759029. [10] A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces, Monographs and Textbooks in Pure and Applied Mathematics, 215, Marcel Dekker, Inc., New York, 1999. [11] D. Guidetti, On maximal regularity for the Cauchy-Dirichlet mixed parabolic problem with fractional time derivative, Bruno Pini Mathematical Analysis Seminar, 9, Univ. Bologna, Bologna, 2018,147–157. [12] J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Die Grundlehren der Mathematischen Wissenschaften, 2, Springer-Verlag, New York-Heidelberg, 1972. [13] J. L. Lions and J. Peetre, Sur une classe d'espaces d'interpolation, Inst. Hautes Études Sci. Publ. Math., 19 (1964), 5–68. [14] A. Lorenzi, An Introduction to Identification Problems Via Functional Analysis, Inverse and Ill-Posed Problems Series, Utrecht, Netherlands, 2001. doi: 10.1017/CBO9781139168724. [15] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland Mathematical Library, 18, North-Holland Publishing Co., Amsterdam-New York, 1978.
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