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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

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

show all references

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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