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 HoraniM. FabrizioA. 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. FaviniA. 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. FaviniA. 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.

show all references

References:
[1]

M. Al HoraniM. FabrizioA. 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. FaviniA. 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. FaviniA. 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.

[1]

Zhan-Dong Mei, Jigen Peng, Yang Zhang. On general fractional abstract Cauchy problem. Communications on Pure and Applied Analysis, 2013, 12 (6) : 2753-2772. doi: 10.3934/cpaa.2013.12.2753

[2]

Nguyen Huy Tuan, Mokhtar Kirane, Long Dinh Le, Van Thinh Nguyen. On an inverse problem for fractional evolution equation. Evolution Equations and Control Theory, 2017, 6 (1) : 111-134. doi: 10.3934/eect.2017007

[3]

Hernan R. Henriquez. Generalized solutions for the abstract singular Cauchy problem. Communications on Pure and Applied Analysis, 2009, 8 (3) : 955-976. doi: 10.3934/cpaa.2009.8.955

[4]

Poongodi Rathinasamy, Murugesu Rangasamy, Nirmalkumar Rajendran. Exact controllability results for a class of abstract nonlocal Cauchy problem with impulsive conditions. Evolution Equations and Control Theory, 2017, 6 (4) : 599-613. doi: 10.3934/eect.2017030

[5]

Pengyu Chen, Yongxiang Li, Xuping Zhang. Cauchy problem for stochastic non-autonomous evolution equations governed by noncompact evolution families. Discrete and Continuous Dynamical Systems - B, 2021, 26 (3) : 1531-1547. doi: 10.3934/dcdsb.2020171

[6]

Pengyu Chen, Yongxiang Li, Xuping Zhang. On the initial value problem of fractional stochastic evolution equations in Hilbert spaces. Communications on Pure and Applied Analysis, 2015, 14 (5) : 1817-1840. doi: 10.3934/cpaa.2015.14.1817

[7]

Hiroyuki Hirayama, Mamoru Okamoto. Random data Cauchy problem for the nonlinear Schrödinger equation with derivative nonlinearity. Discrete and Continuous Dynamical Systems, 2016, 36 (12) : 6943-6974. doi: 10.3934/dcds.2016102

[8]

Phan Van Tin. On the Cauchy problem for a derivative nonlinear Schrödinger equation with nonvanishing boundary conditions. Evolution Equations and Control Theory, 2022, 11 (3) : 837-867. doi: 10.3934/eect.2021028

[9]

Huy Tuan Nguyen, Huu Can Nguyen, Renhai Wang, Yong Zhou. Initial value problem for fractional Volterra integro-differential equations with Caputo derivative. Discrete and Continuous Dynamical Systems - B, 2021, 26 (12) : 6483-6510. doi: 10.3934/dcdsb.2021030

[10]

Christina A. Hollon, Jeffrey T. Neugebauer. Positive solutions of a fractional boundary value problem with a fractional derivative boundary condition. Conference Publications, 2015, 2015 (special) : 615-620. doi: 10.3934/proc.2015.0615

[11]

Eddye Bustamante, José Jiménez Urrea, Jorge Mejía. The Cauchy problem for a family of two-dimensional fractional Benjamin-Ono equations. Communications on Pure and Applied Analysis, 2019, 18 (3) : 1177-1203. doi: 10.3934/cpaa.2019057

[12]

Lili Chang, Wei Gong, Guiquan Sun, Ningning Yan. PDE-constrained optimal control approach for the approximation of an inverse Cauchy problem. Inverse Problems and Imaging, 2015, 9 (3) : 791-814. doi: 10.3934/ipi.2015.9.791

[13]

Li Liang. Increasing stability for the inverse problem of the Schrödinger equation with the partial Cauchy data. Inverse Problems and Imaging, 2015, 9 (2) : 469-478. doi: 10.3934/ipi.2015.9.469

[14]

Abderrahmane Habbal, Moez Kallel, Marwa Ouni. Nash strategies for the inverse inclusion Cauchy-Stokes problem. Inverse Problems and Imaging, 2019, 13 (4) : 827-862. doi: 10.3934/ipi.2019038

[15]

Masahiro Yamamoto. Uniqueness for inverse problem of determining fractional orders for time-fractional advection-diffusion equations. Mathematical Control and Related Fields, 2022  doi: 10.3934/mcrf.2022017

[16]

Felipe Linares, M. Panthee. On the Cauchy problem for a coupled system of KdV equations. Communications on Pure and Applied Analysis, 2004, 3 (3) : 417-431. doi: 10.3934/cpaa.2004.3.417

[17]

Paolo Antonelli, Daniel Marahrens, Christof Sparber. On the Cauchy problem for nonlinear Schrödinger equations with rotation. Discrete and Continuous Dynamical Systems, 2012, 32 (3) : 703-715. doi: 10.3934/dcds.2012.32.703

[18]

Ahmad Z. Fino, Mokhtar Kirane. The Cauchy problem for heat equation with fractional Laplacian and exponential nonlinearity. Communications on Pure and Applied Analysis, 2020, 19 (7) : 3625-3650. doi: 10.3934/cpaa.2020160

[19]

Shuai Zhang, Shaopeng Xu. The probabilistic Cauchy problem for the fourth order Schrödinger equation with special derivative nonlinearities. Communications on Pure and Applied Analysis, 2020, 19 (6) : 3367-3385. doi: 10.3934/cpaa.2020149

[20]

Editorial Office. Retraction: The probabilistic Cauchy problem for the fourth order Schrödinger equation with special derivative nonlinearities. Communications on Pure and Applied Analysis, 2020, 19 (7) : 3785-3785. doi: 10.3934/cpaa.2020167

2020 Impact Factor: 2.425

Metrics

  • PDF downloads (358)
  • HTML views (262)
  • Cited by (1)

[Back to Top]