# 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  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 & 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.  Google Scholar [2] E. G. Bazhlekova, Fractional evolution equations in Banach spaces, Dissertation, Eindhoven University of Technology in Eindhoven, 2001.  Google Scholar [3] A. Favaron, A. Favini and H. Tanabe, Perturbation methods for inverse problems on degenerate differential equations, preprint. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [7] A. Favini, A. Lorenzi and H. Tanabe, Singular integro-differential equations of parabolic type, Adv. Differential Equations, 7 (2002), 769-798.   Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [13] J. L. Lions and J. Peetre, Sur une classe d'espaces d'interpolation, Inst. Hautes Études Sci. Publ. Math., 19 (1964), 5–68.  Google Scholar [14] A. Lorenzi, An Introduction to Identification Problems Via Functional Analysis, Inverse and Ill-Posed Problems Series, Utrecht, Netherlands, 2001. doi: 10.1017/CBO9781139168724.  Google Scholar [15] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland Mathematical Library, 18, North-Holland Publishing Co., Amsterdam-New York, 1978.  Google Scholar

show all references

##### 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.  Google Scholar [2] E. G. Bazhlekova, Fractional evolution equations in Banach spaces, Dissertation, Eindhoven University of Technology in Eindhoven, 2001.  Google Scholar [3] A. Favaron, A. Favini and H. Tanabe, Perturbation methods for inverse problems on degenerate differential equations, preprint. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [7] A. Favini, A. Lorenzi and H. Tanabe, Singular integro-differential equations of parabolic type, Adv. Differential Equations, 7 (2002), 769-798.   Google Scholar [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. Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [13] J. L. Lions and J. Peetre, Sur une classe d'espaces d'interpolation, Inst. Hautes Études Sci. Publ. Math., 19 (1964), 5–68.  Google Scholar [14] A. Lorenzi, An Introduction to Identification Problems Via Functional Analysis, Inverse and Ill-Posed Problems Series, Utrecht, Netherlands, 2001. doi: 10.1017/CBO9781139168724.  Google Scholar [15] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland Mathematical Library, 18, North-Holland Publishing Co., Amsterdam-New York, 1978.  Google Scholar
 [1] Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348 [2] Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380 [3] Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056 [4] Min Chen, Olivier Goubet, Shenghao Li. Mathematical analysis of bump to bucket problem. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5567-5580. doi: 10.3934/cpaa.2020251 [5] Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253 [6] Pengyu Chen. Non-autonomous stochastic evolution equations with nonlinear noise and nonlocal conditions governed by noncompact evolution families. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020383 [7] Stefano Bianchini, Paolo Bonicatto. Forward untangling and applications to the uniqueness problem for the continuity equation. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020384 [8] S. Sadeghi, H. Jafari, S. Nemati. Solving fractional Advection-diffusion equation using Genocchi operational matrix based on Atangana-Baleanu derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020435 [9] Marco Ghimenti, Anna Maria Micheletti. Compactness results for linearly perturbed Yamabe problem on manifolds with boundary. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020453 [10] Alberto Bressan, Sondre Tesdal Galtung. A 2-dimensional shape optimization problem for tree branches. Networks & Heterogeneous Media, 2020  doi: 10.3934/nhm.2020031 [11] Fioralba Cakoni, Pu-Zhao Kow, Jenn-Nan Wang. The interior transmission eigenvalue problem for elastic waves in media with obstacles. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020075 [12] Shun Zhang, Jianlin Jiang, Su Zhang, Yibing Lv, Yuzhen Guo. ADMM-type methods for generalized multi-facility Weber problem. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020171 [13] Shenglan Xie, Maoan Han, Peng Zhu. A posteriori error estimate of weak Galerkin fem for second order elliptic problem with mixed boundary condition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020340 [14] Gang Bao, Mingming Zhang, Bin Hu, Peijun Li. An adaptive finite element DtN method for the three-dimensional acoustic scattering problem. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020351 [15] Mehdi Badsi. Collisional sheath solutions of a bi-species Vlasov-Poisson-Boltzmann boundary value problem. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020052 [16] Lingfeng Li, Shousheng Luo, Xue-Cheng Tai, Jiang Yang. A new variational approach based on level-set function for convex hull problem with outliers. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020070 [17] Serena Dipierro, Benedetta Pellacci, Enrico Valdinoci, Gianmaria Verzini. Time-fractional equations with reaction terms: Fundamental solutions and asymptotics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 257-275. doi: 10.3934/dcds.2020137 [18] Bahaaeldin Abdalla, Thabet Abdeljawad. Oscillation criteria for kernel function dependent fractional dynamic equations. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020443 [19] Hoang The Tuan. On the asymptotic behavior of solutions to time-fractional elliptic equations driven by a multiplicative white noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020318 [20] Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049

2019 Impact Factor: 1.233