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.

show all references

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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