Inverse problems for evolution equations with time dependent operator-coefficients

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

Inverse problems for evolution equations with time dependent operator-coefficients

Mohammed Al Horani ^1, , Angelo Favini ^2, and Hiroki Tanabe ^3,

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

Abstract
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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.

Keywords: inverse problem, Banach space., degenerate equation, first order equations, Identification problem.

Mathematics Subject Classification: Primary: 34G10; Secondary: 34A5.

Citation: Mohammed Al Horani, Angelo Favini, Hiroki Tanabe. Inverse problems for evolution equations with time dependent operator-coefficients. Discrete & 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. Google Scholar
[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), 251 (2006), 1. doi: 10.1201/9781420011135.ch1. Google Scholar
[3]	M. Al Horani and A. Favini, An identification problem for first-order degenerate differential equations,, J. Optim. Theory Appl., 130* (2006), 41. doi: 10.1007/s10957-006-9083-y. Google Scholar*
[4]	M. Al Horani and A. Favini, Degenerate first-order inverse problems in Banach spaces,, Nonlinear Anal., 75 (2012), 68. doi: 10.1016/j.na.2011.08.001. Google Scholar
[5]	M. Al Horani and A. Favini, First-order inverse evolution equations,, Evol. Equ. Control Theory, 3 (2014), 355. doi: 10.3934/eect.2014.3.355. Google Scholar
[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. doi: 10.3934/dcdsb.2014.19.2159. Google Scholar
[7]	M. Al Horani and A. Favini, Perturbation method for first and complete second order differential equations,, J. Optim. Theory Appl., 166 (2015), 949. doi: 10.1007/s10957-015-0733-9. Google Scholar
[8]	H. Amann, Linear and Quasilinear Parabolic Problems. Vol. I. Abstract Linear Theory,, Monographs in Mathematics, 89 (1995). doi: 10.1007/978-3-0348-9221-6. Google Scholar
[9]	S. Bertoni, Stability of CD-systems under perturbations in the Favard class,, Mediterr. J. Math., 11 (2014), 1195. doi: 10.1007/s00009-013-0376-8. Google Scholar
[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. Google Scholar
[11]	A. Favini and A. Lorenzi, Identification problems in singular integro-differential equations of parabolic type II,, Nonlinear Anal., 56 (2004), 879. doi: 10.1016/j.na.2003.10.018. Google Scholar
[12]	A. Favini, A. Lorenzi, G. Marinoschi and H. Tanabe, Perturbation methods and identification problems for degenerate evolution systems,, Advances in mathematics, (2013), 145. Google Scholar
[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. Google Scholar
[14]	A. Favini, A. Lorenzi and H. Tanabe, First-order regular and degenerate identification differential problems,, Abstr. Appl. Anal., (2015). doi: 10.1155/2015/393624. Google Scholar
[15]	A. Favini, A. Lorenzi and H. Tanabe, Degenerate integro-differential equations of parabolic type with Robin boundary conditions,, submitted., (). Google Scholar
[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. Google Scholar
[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. doi: 10.1007/s10957-009-9635-z. Google Scholar
[18]	A. Favini and H. Tanabe, Degenerate differential equations of parabolic type and inverse problems,, Proceeding, (2015), 89. Google Scholar
[19]	A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces,, Marcel Dekker. Inc. New York, (1999). Google Scholar
[20]	A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems,, Birkhäuser, (1995). doi: 10.1007/978-3-0348-9234-6. Google Scholar
[21]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differntial Equations,, Applied Mathematical Sciences 44, 44 (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar
[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). Google Scholar
[23]	H, Tanabe, Functional Analytic Methods for Partial Differential Equations,, Monographs and Textbooks in Pure and Applied Mathematics 204, 204 (1997). Google Scholar

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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. Google Scholar
[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), 251 (2006), 1. doi: 10.1201/9781420011135.ch1. Google Scholar
[3]	M. Al Horani and A. Favini, An identification problem for first-order degenerate differential equations,, J. Optim. Theory Appl., 130* (2006), 41. doi: 10.1007/s10957-006-9083-y. Google Scholar*
[4]	M. Al Horani and A. Favini, Degenerate first-order inverse problems in Banach spaces,, Nonlinear Anal., 75 (2012), 68. doi: 10.1016/j.na.2011.08.001. Google Scholar
[5]	M. Al Horani and A. Favini, First-order inverse evolution equations,, Evol. Equ. Control Theory, 3 (2014), 355. doi: 10.3934/eect.2014.3.355. Google Scholar
[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. doi: 10.3934/dcdsb.2014.19.2159. Google Scholar
[7]	M. Al Horani and A. Favini, Perturbation method for first and complete second order differential equations,, J. Optim. Theory Appl., 166 (2015), 949. doi: 10.1007/s10957-015-0733-9. Google Scholar
[8]	H. Amann, Linear and Quasilinear Parabolic Problems. Vol. I. Abstract Linear Theory,, Monographs in Mathematics, 89 (1995). doi: 10.1007/978-3-0348-9221-6. Google Scholar
[9]	S. Bertoni, Stability of CD-systems under perturbations in the Favard class,, Mediterr. J. Math., 11 (2014), 1195. doi: 10.1007/s00009-013-0376-8. Google Scholar
[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. Google Scholar
[11]	A. Favini and A. Lorenzi, Identification problems in singular integro-differential equations of parabolic type II,, Nonlinear Anal., 56 (2004), 879. doi: 10.1016/j.na.2003.10.018. Google Scholar
[12]	A. Favini, A. Lorenzi, G. Marinoschi and H. Tanabe, Perturbation methods and identification problems for degenerate evolution systems,, Advances in mathematics, (2013), 145. Google Scholar
[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. Google Scholar
[14]	A. Favini, A. Lorenzi and H. Tanabe, First-order regular and degenerate identification differential problems,, Abstr. Appl. Anal., (2015). doi: 10.1155/2015/393624. Google Scholar
[15]	A. Favini, A. Lorenzi and H. Tanabe, Degenerate integro-differential equations of parabolic type with Robin boundary conditions,, submitted., (). Google Scholar
[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. Google Scholar
[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. doi: 10.1007/s10957-009-9635-z. Google Scholar
[18]	A. Favini and H. Tanabe, Degenerate differential equations of parabolic type and inverse problems,, Proceeding, (2015), 89. Google Scholar
[19]	A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces,, Marcel Dekker. Inc. New York, (1999). Google Scholar
[20]	A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems,, Birkhäuser, (1995). doi: 10.1007/978-3-0348-9234-6. Google Scholar
[21]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differntial Equations,, Applied Mathematical Sciences 44, 44 (1983). doi: 10.1007/978-1-4612-5561-1. Google Scholar
[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). Google Scholar
[23]	H, Tanabe, Functional Analytic Methods for Partial Differential Equations,, Monographs and Textbooks in Pure and Applied Mathematics 204, 204 (1997). Google Scholar

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