December  2020, 13(12): 3285-3304. doi: 10.3934/dcdss.2020240

Fractional Cauchy problems for infinite interval case

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

* Corresponding author: Mohammed Al Horani

Received  December 2018 Revised  October 2019 Published  January 2020

We are devoted with 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 from 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 for infinite interval case. Discrete & Continuous Dynamical Systems - S, 2020, 13 (12) : 3285-3304. doi: 10.3934/dcdss.2020240
References:
[1]

M. Al HoraniA. Favini and H. Tanabe, Direct and inverse fractional abstract Cauchy problems, Mathematics, 7 (2019), 1-9.   Google Scholar

[2]

M. Al Horani, M. Fabrizio, A. Favini and H. Tanabe, Fractional Cauchy problems and applications, Discrete & Continuous Dynamical Systems-Series S, to appear. Google Scholar

[3]

M. Al HoraniM. FabrizioA. Favini and H. Tanabe, Direct and inverse problems for degenerate differential equations, Ann. Univ. Ferrara, 64 (2018), 227-241.  doi: 10.1007/s11565-018-0303-9.  Google Scholar

[4]

E. G. Bazhlekova,, Fractional Evolution Equations in Banach Spaces, Eindhoven University of Technology, 2001.  Google Scholar

[5]

A. Favaron, A. Favini and H. Tanabe, Perturbation methods for inverse problems on degenerate differential equations, preprint. Google Scholar

[6]

A. Favini, A. Lorenzi, G. Marinoschi and H. Tanabe, Perturbation methods and identification problems for degenerate evolution systems, Advances in Mathematics, Ed. Acad. Române, Bucharest, (2013), 145-156.  Google Scholar

[7]

A. FaviniA. Lorenzi and H. Tanabe, Degenerate integro-differential equations of parabolic type with Robin boundary conditions, Journal of Mathematical Analysis and Applications, 447 (2017), 579-665.  doi: 10.1016/j.jmaa.2016.10.029.  Google Scholar

[8]

A. FaviniA. Lorenzi and H. Tanabe, Direct and inverse degenerate parabolic differential equations with multi-valued operators, Electron. J. Diff. Equ., 2015 (2015), 1-22.   Google Scholar

[9]

A. FaviniA. Lorenzi and H. Tanabe, Singular integro-differential equations of parabolic type, Advances in Differential Equations, 7 (2002), 769-798.   Google Scholar

[10]

A. Favini and H. Tanabe, Degenerate differential equations of parabolic type and inverse problems, Proceeding, Seminar on Partial Differential Equations, Osaka University, Osaka (2015), 89-100. Google Scholar

[11]

A. Favini and A. Yagi, Multivalued linear operators and degenerate evolution equations, Annali Di Matematica Pura ed Applicata, 163 (1993), 353-384.  doi: 10.1007/BF01759029.  Google Scholar

[12]

A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces, Marcel Dekker. Inc. New York, 1999.  Google Scholar

[13]

V. Fedorov and N. D. Ivanova, Identification problem for degenerate evolution equations of fractional order, Fractional Calculus and Applied Analysis, 20 (2017), 706-721.   Google Scholar

[14]

D. Guidetti, On maximal regularity for the Cauchy-Dirichlet mixed parabolic problem with fractional time derivative, Bruno Pini Mathematical Analysis Seminar, 9 (2018), 147-157.   Google Scholar

[15]

T. Kato, Fractional powers of dissipative operators, J. Math. Soc. Japan, 13 (1961), 246-274.  doi: 10.2969/jmsj/01330246.  Google Scholar

[16]

J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems And Applications, Springer-Verlag, Berlin, 1972.  Google Scholar

[17]

J. L. Lions and J. Peetre, Sur une classe d'espaces d'interpolation, Publications Mathmatiques de l'IHES, 19 (1964), 5-68.   Google Scholar

[18]

A. Lorenzi, An Introduction to Identification Problems Via Functional Analysis, VSP, Utrecht, The Netherland, 2001. Google Scholar

[19]

G. A. Sviridyuk and V. E. Fedorov, Linear Sobolev Type Equations and Degenerate Semigroups of Operators, VSP, Utrecht, Boston, 2003.  Google Scholar

[20]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amesterdam, 1978.  Google Scholar

show all references

References:
[1]

M. Al HoraniA. Favini and H. Tanabe, Direct and inverse fractional abstract Cauchy problems, Mathematics, 7 (2019), 1-9.   Google Scholar

[2]

M. Al Horani, M. Fabrizio, A. Favini and H. Tanabe, Fractional Cauchy problems and applications, Discrete & Continuous Dynamical Systems-Series S, to appear. Google Scholar

[3]

M. Al HoraniM. FabrizioA. Favini and H. Tanabe, Direct and inverse problems for degenerate differential equations, Ann. Univ. Ferrara, 64 (2018), 227-241.  doi: 10.1007/s11565-018-0303-9.  Google Scholar

[4]

E. G. Bazhlekova,, Fractional Evolution Equations in Banach Spaces, Eindhoven University of Technology, 2001.  Google Scholar

[5]

A. Favaron, A. Favini and H. Tanabe, Perturbation methods for inverse problems on degenerate differential equations, preprint. Google Scholar

[6]

A. Favini, A. Lorenzi, G. Marinoschi and H. Tanabe, Perturbation methods and identification problems for degenerate evolution systems, Advances in Mathematics, Ed. Acad. Române, Bucharest, (2013), 145-156.  Google Scholar

[7]

A. FaviniA. Lorenzi and H. Tanabe, Degenerate integro-differential equations of parabolic type with Robin boundary conditions, Journal of Mathematical Analysis and Applications, 447 (2017), 579-665.  doi: 10.1016/j.jmaa.2016.10.029.  Google Scholar

[8]

A. FaviniA. Lorenzi and H. Tanabe, Direct and inverse degenerate parabolic differential equations with multi-valued operators, Electron. J. Diff. Equ., 2015 (2015), 1-22.   Google Scholar

[9]

A. FaviniA. Lorenzi and H. Tanabe, Singular integro-differential equations of parabolic type, Advances in Differential Equations, 7 (2002), 769-798.   Google Scholar

[10]

A. Favini and H. Tanabe, Degenerate differential equations of parabolic type and inverse problems, Proceeding, Seminar on Partial Differential Equations, Osaka University, Osaka (2015), 89-100. Google Scholar

[11]

A. Favini and A. Yagi, Multivalued linear operators and degenerate evolution equations, Annali Di Matematica Pura ed Applicata, 163 (1993), 353-384.  doi: 10.1007/BF01759029.  Google Scholar

[12]

A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces, Marcel Dekker. Inc. New York, 1999.  Google Scholar

[13]

V. Fedorov and N. D. Ivanova, Identification problem for degenerate evolution equations of fractional order, Fractional Calculus and Applied Analysis, 20 (2017), 706-721.   Google Scholar

[14]

D. Guidetti, On maximal regularity for the Cauchy-Dirichlet mixed parabolic problem with fractional time derivative, Bruno Pini Mathematical Analysis Seminar, 9 (2018), 147-157.   Google Scholar

[15]

T. Kato, Fractional powers of dissipative operators, J. Math. Soc. Japan, 13 (1961), 246-274.  doi: 10.2969/jmsj/01330246.  Google Scholar

[16]

J. L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems And Applications, Springer-Verlag, Berlin, 1972.  Google Scholar

[17]

J. L. Lions and J. Peetre, Sur une classe d'espaces d'interpolation, Publications Mathmatiques de l'IHES, 19 (1964), 5-68.   Google Scholar

[18]

A. Lorenzi, An Introduction to Identification Problems Via Functional Analysis, VSP, Utrecht, The Netherland, 2001. Google Scholar

[19]

G. A. Sviridyuk and V. E. Fedorov, Linear Sobolev Type Equations and Degenerate Semigroups of Operators, VSP, Utrecht, Boston, 2003.  Google Scholar

[20]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amesterdam, 1978.  Google Scholar

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