# American Institute of Mathematical Sciences

June  2016, 9(3): 737-744. doi: 10.3934/dcdss.2016025

## Inverse problems for evolution equations with time dependent operator-coefficients

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

Received  May 2015 Revised  July 2015 Published  April 2016

In this paper we study an inverse problem with time dependent operator-coefficients. We indicate sufficient conditions for the existence and the uniqueness of a solution to this problem. A number of concrete applications to partial differential equations is described.
Citation: Mohammed Al Horani, Angelo Favini, Hiroki Tanabe. Inverse problems for evolution equations with time dependent operator-coefficients. Discrete and Continuous Dynamical Systems - S, 2016, 9 (3) : 737-744. doi: 10.3934/dcdss.2016025
##### References:
 [1] P. Acquistapace, A unified approach to abstract linear nonautonomous parabolic equations, Rend. Sem. Mat.Univ. Padova, 78 (1987), 47-107. [2] M. Al Horani and A. Favini, Degenerate first-order identification problems in Banach spaces, in Differential equations: inverse and direct problems (eds. A. Favini and A. Lorenzi), Taylor and Francis Group, 251 (2006), 1-15. doi: 10.1201/9781420011135.ch1. [3] M. Al Horani and A. Favini, An identification problem for first-order degenerate differential equations, J. Optim. Theory Appl., 130 (2006), 41-60. doi: 10.1007/s10957-006-9083-y. [4] M. Al Horani and A. Favini, Degenerate first-order inverse problems in Banach spaces, Nonlinear Anal., 75 (2012), 68-77. doi: 10.1016/j.na.2011.08.001. [5] M. Al Horani and A. Favini, First-order inverse evolution equations, Evol. Equ. Control Theory, 3 (2014), 355-361. doi: 10.3934/eect.2014.3.355. [6] M. Al Horani and A. Favini, Inverse problems for singular differential-operator equations with higher order polar singularities, Discrete. Contin. Dyn. Syst. Ser. B, 19 (2014), 2159-2168. doi: 10.3934/dcdsb.2014.19.2159. [7] M. Al Horani and A. Favini, Perturbation method for first and complete second order differential equations, J. Optim. Theory Appl., 166 (2015), 949-967. doi: 10.1007/s10957-015-0733-9. [8] H. Amann, Linear and Quasilinear Parabolic Problems. Vol. I. Abstract Linear Theory, Monographs in Mathematics, 89, Birkhauser, Basel-Borton-Berlin, 1995. doi: 10.1007/978-3-0348-9221-6. [9] S. Bertoni, Stability of CD-systems under perturbations in the Favard class, Mediterr. J. Math., 11 (2014), 1195-1204. doi: 10.1007/s00009-013-0376-8. [10] A. Favini and A. Lorenzi, Identification problems for singular integro-differential equations of parabolic type I, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 12 (2005), 303-328. [11] A. Favini and A. Lorenzi, Identification problems in singular integro-differential equations of parabolic type II, Nonlinear Anal., 56 (2004), 879-904. doi: 10.1016/j.na.2003.10.018. [12] 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. [13] A. Favini, A. Lorenzi and H. Tanabe, Direct and inverse problems for systems of singular differential boundary-value problems, Electron. J. Differential Equations, 225 (2012), 1-34. [14] A. Favini, A. Lorenzi and H. Tanabe, First-order regular and degenerate identification differential problems, Abstr. Appl. Anal., (2015), Art. ID 393624, 42 pp. doi: 10.1155/2015/393624. [15] A. Favini, A. Lorenzi and H. Tanabe, Degenerate integro-differential equations of parabolic type with Robin boundary conditions, submitted. [16] A. Favini, A. Lorenzi and H. Tanabe, Direct and inverse degenerate parabolic differential equations with multi-valued operators, Electron. J. Diff. Equ., 2015 (2015), 1-22. [17] A. Favini and G. Marinoschi, Identification of the time derivative coefficient in a fast diffusion degenerate equation, J. Optim. Theory Appl., 145 (2010), 249-269. doi: 10.1007/s10957-009-9635-z. [18] 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. [19] A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces, Marcel Dekker. Inc. New York, 1999. [20] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, Basel, 1995. doi: 10.1007/978-3-0348-9234-6. [21] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differntial Equations, Applied Mathematical Sciences 44, Springer, Berlin, 1983. doi: 10.1007/978-1-4612-5561-1. [22] A.I. Prilepko, D.G. Orlovsky and I.A. Vasin, Methods for Solving Inverse Problems in Mathematical Physics, Marcel Dekker. Inc. New York, 2000. [23] H, Tanabe, Functional Analytic Methods for Partial Differential Equations, Monographs and Textbooks in Pure and Applied Mathematics 204, Marcel Dekker, Inc. New York, 1997.

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##### References:
 [1] P. Acquistapace, A unified approach to abstract linear nonautonomous parabolic equations, Rend. Sem. Mat.Univ. Padova, 78 (1987), 47-107. [2] M. Al Horani and A. Favini, Degenerate first-order identification problems in Banach spaces, in Differential equations: inverse and direct problems (eds. A. Favini and A. Lorenzi), Taylor and Francis Group, 251 (2006), 1-15. doi: 10.1201/9781420011135.ch1. [3] M. Al Horani and A. Favini, An identification problem for first-order degenerate differential equations, J. Optim. Theory Appl., 130 (2006), 41-60. doi: 10.1007/s10957-006-9083-y. [4] M. Al Horani and A. Favini, Degenerate first-order inverse problems in Banach spaces, Nonlinear Anal., 75 (2012), 68-77. doi: 10.1016/j.na.2011.08.001. [5] M. Al Horani and A. Favini, First-order inverse evolution equations, Evol. Equ. Control Theory, 3 (2014), 355-361. doi: 10.3934/eect.2014.3.355. [6] M. Al Horani and A. Favini, Inverse problems for singular differential-operator equations with higher order polar singularities, Discrete. Contin. Dyn. Syst. Ser. B, 19 (2014), 2159-2168. doi: 10.3934/dcdsb.2014.19.2159. [7] M. Al Horani and A. Favini, Perturbation method for first and complete second order differential equations, J. Optim. Theory Appl., 166 (2015), 949-967. doi: 10.1007/s10957-015-0733-9. [8] H. Amann, Linear and Quasilinear Parabolic Problems. Vol. I. Abstract Linear Theory, Monographs in Mathematics, 89, Birkhauser, Basel-Borton-Berlin, 1995. doi: 10.1007/978-3-0348-9221-6. [9] S. Bertoni, Stability of CD-systems under perturbations in the Favard class, Mediterr. J. Math., 11 (2014), 1195-1204. doi: 10.1007/s00009-013-0376-8. [10] A. Favini and A. Lorenzi, Identification problems for singular integro-differential equations of parabolic type I, Dyn. Contin. Discrete Impuls. Syst. Ser. A Math. Anal., 12 (2005), 303-328. [11] A. Favini and A. Lorenzi, Identification problems in singular integro-differential equations of parabolic type II, Nonlinear Anal., 56 (2004), 879-904. doi: 10.1016/j.na.2003.10.018. [12] 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. [13] A. Favini, A. Lorenzi and H. Tanabe, Direct and inverse problems for systems of singular differential boundary-value problems, Electron. J. Differential Equations, 225 (2012), 1-34. [14] A. Favini, A. Lorenzi and H. Tanabe, First-order regular and degenerate identification differential problems, Abstr. Appl. Anal., (2015), Art. ID 393624, 42 pp. doi: 10.1155/2015/393624. [15] A. Favini, A. Lorenzi and H. Tanabe, Degenerate integro-differential equations of parabolic type with Robin boundary conditions, submitted. [16] A. Favini, A. Lorenzi and H. Tanabe, Direct and inverse degenerate parabolic differential equations with multi-valued operators, Electron. J. Diff. Equ., 2015 (2015), 1-22. [17] A. Favini and G. Marinoschi, Identification of the time derivative coefficient in a fast diffusion degenerate equation, J. Optim. Theory Appl., 145 (2010), 249-269. doi: 10.1007/s10957-009-9635-z. [18] 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. [19] A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces, Marcel Dekker. Inc. New York, 1999. [20] A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, Basel, 1995. doi: 10.1007/978-3-0348-9234-6. [21] A. Pazy, Semigroups of Linear Operators and Applications to Partial Differntial Equations, Applied Mathematical Sciences 44, Springer, Berlin, 1983. doi: 10.1007/978-1-4612-5561-1. [22] A.I. Prilepko, D.G. Orlovsky and I.A. Vasin, Methods for Solving Inverse Problems in Mathematical Physics, Marcel Dekker. Inc. New York, 2000. [23] H, Tanabe, Functional Analytic Methods for Partial Differential Equations, Monographs and Textbooks in Pure and Applied Mathematics 204, Marcel Dekker, Inc. New York, 1997.
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