September  2014, 3(3): 355-361. doi: 10.3934/eect.2014.3.355

First-order inverse evolution equations

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

Received  May 2013 Revised  April 2014 Published  August 2014

We are concerned with an inverse problem for first order linear evolution equations. We indicate sufficient conditions for existence and uniqueness of a solution to these problems. All the results apply well to inverse problems for equations from mathematical physics. Indeed, as a possible application of the abstract theorems, some examples of partial differential equations are given.
Citation: Mohammed Al Horani, Angelo Favini. First-order inverse evolution equations. Evolution Equations and Control Theory, 2014, 3 (3) : 355-361. doi: 10.3934/eect.2014.3.355
References:
[1]

M. Al Horani, An identification problem for some degenerate differential equations, Matematiche (Catania), 57 (2002), 217-227.

[2]

M. Al Horani, Projection method for solving degenerate first-order identification problem, J. Math. Anal. Appl., 364 (2010), 204-208. doi: 10.1016/j.jmaa.2009.10.033.

[3]

M. Al Horani and A. Favini, An identification problem for first-order degenerate differential equations, Journal of Optimization Theory and Applications, 130 (2006), 41-60. doi: 10.1007/s10957-006-9083-y.

[4]

M. Al Horani and A. Favini, Degenerate first-order identification problems in Banach spaces, Differential Equations, Inverse and Direct Problems, Taylor and Francis Group, Boca Raton (eds. A. Favini and A. Lorenzi), 1-15. doi: 10.1201/9781420011135.ch1.

[5]

M. Al Horani and A. Favini, Perturbation method for first and complete second order differential equations, Preprint.

[6]

W. Desch and W. Schappacher, On relatively bounded perturbations of linear $C_0$-semigroups, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 11 (1984), 327-341.

[7]

N. Dunford and T. Schwarz, Linear Operators, I, Wiley (Interscience), New York, 1958.

[8]

K.-J. Engel and R. Nagel, One Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Math., Springer-Verlag, Berlin-Heidelgerg-New York, 2000.

[9]

A. Favini and A. Lorenzi, Identification problems for singular integro-differential equations of parabolic type I, Dynamics of Continuous, Discrete and Impulsive Systems, Series A; Mathematical Analysis, 12 (2005), 303-328.

[10]

A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces, Marcel Dekker. Inc., New York, 1999.

[11]

A. Favini and S. Romanelli, Analicity semigroups on $C[0,1]$ generated by some classes of second order differential equations, Semigroup Forum, 56 (1998), 362-372.

[12]

G. Greiner, Spectral properties and asymptotic behavior of the linear transport equation, Math. Zeit., 185 (1984), 167-177. doi: 10.1007/BF01181687.

[13]

A. Lorenzi, An Introduction to Identification Problems Via Functional Analysis, VSP, Utrecht, The Netherland, 2001. doi: 10.1515/9783110940923.

[14]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, $1^{ist}$ ed, Birkhäuser, Basel, 1995.

[15]

I. Prilepko, G. Orlovsky and A. Vasin, Methods for Solving Inverse Problems in Mathematical Physics, Marcel Dekker. Inc., New York, 2000.

[16]

E. Sinestrari, Wave equation with memory, Discrete Contin. Dynam. Systems, 5 (1999), 881-896. doi: 10.3934/dcds.1999.5.881.

[17]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amesterdam, 1978.

show all references

References:
[1]

M. Al Horani, An identification problem for some degenerate differential equations, Matematiche (Catania), 57 (2002), 217-227.

[2]

M. Al Horani, Projection method for solving degenerate first-order identification problem, J. Math. Anal. Appl., 364 (2010), 204-208. doi: 10.1016/j.jmaa.2009.10.033.

[3]

M. Al Horani and A. Favini, An identification problem for first-order degenerate differential equations, Journal of Optimization Theory and Applications, 130 (2006), 41-60. doi: 10.1007/s10957-006-9083-y.

[4]

M. Al Horani and A. Favini, Degenerate first-order identification problems in Banach spaces, Differential Equations, Inverse and Direct Problems, Taylor and Francis Group, Boca Raton (eds. A. Favini and A. Lorenzi), 1-15. doi: 10.1201/9781420011135.ch1.

[5]

M. Al Horani and A. Favini, Perturbation method for first and complete second order differential equations, Preprint.

[6]

W. Desch and W. Schappacher, On relatively bounded perturbations of linear $C_0$-semigroups, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 11 (1984), 327-341.

[7]

N. Dunford and T. Schwarz, Linear Operators, I, Wiley (Interscience), New York, 1958.

[8]

K.-J. Engel and R. Nagel, One Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Math., Springer-Verlag, Berlin-Heidelgerg-New York, 2000.

[9]

A. Favini and A. Lorenzi, Identification problems for singular integro-differential equations of parabolic type I, Dynamics of Continuous, Discrete and Impulsive Systems, Series A; Mathematical Analysis, 12 (2005), 303-328.

[10]

A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces, Marcel Dekker. Inc., New York, 1999.

[11]

A. Favini and S. Romanelli, Analicity semigroups on $C[0,1]$ generated by some classes of second order differential equations, Semigroup Forum, 56 (1998), 362-372.

[12]

G. Greiner, Spectral properties and asymptotic behavior of the linear transport equation, Math. Zeit., 185 (1984), 167-177. doi: 10.1007/BF01181687.

[13]

A. Lorenzi, An Introduction to Identification Problems Via Functional Analysis, VSP, Utrecht, The Netherland, 2001. doi: 10.1515/9783110940923.

[14]

A. Lunardi, Analytic Semigroups and Optimal Regularity in Parabolic Problems, $1^{ist}$ ed, Birkhäuser, Basel, 1995.

[15]

I. Prilepko, G. Orlovsky and A. Vasin, Methods for Solving Inverse Problems in Mathematical Physics, Marcel Dekker. Inc., New York, 2000.

[16]

E. Sinestrari, Wave equation with memory, Discrete Contin. Dynam. Systems, 5 (1999), 881-896. doi: 10.3934/dcds.1999.5.881.

[17]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, North-Holland, Amesterdam, 1978.

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