# American Institute of Mathematical Sciences

June  2016, 9(3): 687-696. doi: 10.3934/dcdss.2016022

## Identification problem for a degenerate evolution equation with overdetermination on the solution semigroup kernel

 1 Chelyabinsk State University, Laboratory of Quantum Topology, 129 Kashirin Brothers Str., Chelyabinsk, 454001, Russian Federation 2 Chelyabinsk State University, 129 Kashirin Brothers Str., Chelyabinsk, 454001, Russian Federation

Received  February 2015 Revised  July 2015 Published  April 2016

An identification problem is considered for a degenerate evolution equation with overdetermination on the solution semigroup kernel. Solutions of problems with Cauchy and Showalter conditions on initial values are proved to be existing and unique. Solutions stability estimates are derived. The abstract results are applied to an identification problem for the linearized Oskolkov system of equations. There are considered different degrees of system degeneration with respect to the time derivatives of unknown functions.
Citation: Vladimir E. Fedorov, Natalia D. Ivanova. Identification problem for a degenerate evolution equation with overdetermination on the solution semigroup kernel. Discrete and Continuous Dynamical Systems - S, 2016, 9 (3) : 687-696. doi: 10.3934/dcdss.2016022
##### References:
 [1] N. L. Abasheeva, Determination of a right-hand side term in an operator-differential equation of mixed type, J. Inverse Ill-Posed Probl., 10 (2002), 547-560. doi: 10.1515/jiip.2002.10.6.547. [2] 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. [3] 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. [4] G. V. Demidenko and S. V. Uspenskii, Partial Differential Equations and Systems not Solvable with Respect to the Highest-Order Derivative, Marcel Dekker, New York, Basel, 2003. doi: 10.1201/9780203911433. [5] A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces, Marcel Dekker, New York, 1999. [6] V. E. Fedorov, Linear equations of the Sobolev type with relatively $p$-radial operators, Dokl. Akad. Nauk, 351 (1996), 316-318. [7] V. E. Fedorov, Degenerate strongly continuous semigroups of operators, St. Petersburgh. Math. J., 12 (2001), 471-489. [8] V. E. Fedorov, A generalization of the Hille-Yosida theorem to the case of degenerate semigroups in locally convex spaces, Siberian Math. J., 46 (2005), 333-350. doi: 10.1007/s11202-005-0035-9. [9] V. E. Fedorov, Svoistva psevdoresolvent i usloviya sushchestvovaniya vyrozhdennoi polugruppy operatorov, (Russian) [Pseudoresolvent properties and a degenerate operator semigroup existence conditions], Vestnik Chelyab. gos. universiteta. Matematika. Mekhanika. Informatika, 11 (2009), 12-19, 153. [10] N. D. Ivanova, Inverse problem for a linearized quasi-stationary phase field model with degeneracy, Vestnik Yuzhno-Ural'skogo gos. universiteta. Mat. modelirovanie i programmirovanie, 6 (2013), 128-133. [11] N. D. Ivanova, V. E. Fedorov and K. M. Komarova, Nelineinaya obratnaya zadacha dlya sistemy Oskolkova, linearizovannoy v okrestnosti statsionarnogo resheniya, (Russian) [Nonlinear inverse problem for the Oskolkov system, linearized in a stationary solution neighbourhood], Vestnik Chelyab. gos. universiteta. Matematika. Mechanika. Informatika, 13 (2012), 50-71. [12] A. I. Kozhanov, Lineinye obratnye zadachi dlya odnogo klassa vyrozhdayushchikhsya uravneniy sobolevskogo tipa (Russian) [Linear inverse problem for a class of degenerate Sobolev type equations], Vestnik Yuzhno-Ural'skogo gos. universiteta. Mat. modelirovanie i programmirovanie, 5 (2012), 33-42. [13] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, New York-London-Paris: Gordon and Breach, Science Publishers, 1969. [14] A. P. Oskolkov, Nachal'no-kraevye zadachi dlya uravneniy dvizheniya zhidkostei Kel'vina-Foigta i zhidkostei Oldroita, (Russian) [Initial-boundary value problems for equations of Kelvin-Voight and Oldroyd fluids motion], Trudy Mat. instituta AN SSSR, 179 (1988), 126-164. [15] M. V. Plekhanova and V. E. Fedorov, Optimal'noe Upravlenie Vyrozhdennymi Raspredelennymi Sistemami, (Russian) [Optimal Control for Degenerate Distributed Systems], Publishing Center of South Ural State University, Chelyabinsk, 2013. [16] A. I. Prilepko, D. G. Orlovskiy and I. A. Vasin, Methods for Solving Inverse Problems in Mathematical Physics, Marcel Dekker, New York, 2000. [17] A. G. Sveshnikov, A. B. Al'shin, M. O. Korpusov and Yu. D. Pletner, Lineinye i Nelineinye Uravneniya Sobolevskogo Tipa, (Russian) [Linear and Nonlinear Equations of the Sobolev Type], Fizmatlit, Moscow, 2007. [18] A. V. Urazaeva and V. E. Fedorov, An inverse problem for linear Sobolev type equations, J. Inverse Ill-Posed Probl., 12 (2004), 387-395. doi: 10.1515/1569394042248210. [19] A. V. Urazaeva and V. E. Fedorov, Prediction-control problem for some systems of equations of fluid dynamics, Differ. Equ., 44 (2008), 1147-1156. doi: 10.1134/S0012266108080120. [20] A. V. Urazaeva and V. E. Fedorov, On the well-posedness of the prediction-control problem for some systems of equations, Math. Notes, 85 (2009), 426-436. doi: 10.1134/S0001434609030134. [21] A. V. Urazaeva and V. E. Fedorov, Lineinaya evolutsionnaya obratnaya zadacha dlya uravnenii sobolevskogo tipa, (Russian) [Linear evolutionary inverse problem for Sobolev type equations], in Neklassicheskie uravnenia matematicheskoi fiziki (ed. A.I. Kozhanov), Sobolev Institute of Mathematics of the SB RAS, (2010), 293-310.

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##### References:
 [1] N. L. Abasheeva, Determination of a right-hand side term in an operator-differential equation of mixed type, J. Inverse Ill-Posed Probl., 10 (2002), 547-560. doi: 10.1515/jiip.2002.10.6.547. [2] 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. [3] 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. [4] G. V. Demidenko and S. V. Uspenskii, Partial Differential Equations and Systems not Solvable with Respect to the Highest-Order Derivative, Marcel Dekker, New York, Basel, 2003. doi: 10.1201/9780203911433. [5] A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces, Marcel Dekker, New York, 1999. [6] V. E. Fedorov, Linear equations of the Sobolev type with relatively $p$-radial operators, Dokl. Akad. Nauk, 351 (1996), 316-318. [7] V. E. Fedorov, Degenerate strongly continuous semigroups of operators, St. Petersburgh. Math. J., 12 (2001), 471-489. [8] V. E. Fedorov, A generalization of the Hille-Yosida theorem to the case of degenerate semigroups in locally convex spaces, Siberian Math. J., 46 (2005), 333-350. doi: 10.1007/s11202-005-0035-9. [9] V. E. Fedorov, Svoistva psevdoresolvent i usloviya sushchestvovaniya vyrozhdennoi polugruppy operatorov, (Russian) [Pseudoresolvent properties and a degenerate operator semigroup existence conditions], Vestnik Chelyab. gos. universiteta. Matematika. Mekhanika. Informatika, 11 (2009), 12-19, 153. [10] N. D. Ivanova, Inverse problem for a linearized quasi-stationary phase field model with degeneracy, Vestnik Yuzhno-Ural'skogo gos. universiteta. Mat. modelirovanie i programmirovanie, 6 (2013), 128-133. [11] N. D. Ivanova, V. E. Fedorov and K. M. Komarova, Nelineinaya obratnaya zadacha dlya sistemy Oskolkova, linearizovannoy v okrestnosti statsionarnogo resheniya, (Russian) [Nonlinear inverse problem for the Oskolkov system, linearized in a stationary solution neighbourhood], Vestnik Chelyab. gos. universiteta. Matematika. Mechanika. Informatika, 13 (2012), 50-71. [12] A. I. Kozhanov, Lineinye obratnye zadachi dlya odnogo klassa vyrozhdayushchikhsya uravneniy sobolevskogo tipa (Russian) [Linear inverse problem for a class of degenerate Sobolev type equations], Vestnik Yuzhno-Ural'skogo gos. universiteta. Mat. modelirovanie i programmirovanie, 5 (2012), 33-42. [13] O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow, New York-London-Paris: Gordon and Breach, Science Publishers, 1969. [14] A. P. Oskolkov, Nachal'no-kraevye zadachi dlya uravneniy dvizheniya zhidkostei Kel'vina-Foigta i zhidkostei Oldroita, (Russian) [Initial-boundary value problems for equations of Kelvin-Voight and Oldroyd fluids motion], Trudy Mat. instituta AN SSSR, 179 (1988), 126-164. [15] M. V. Plekhanova and V. E. Fedorov, Optimal'noe Upravlenie Vyrozhdennymi Raspredelennymi Sistemami, (Russian) [Optimal Control for Degenerate Distributed Systems], Publishing Center of South Ural State University, Chelyabinsk, 2013. [16] A. I. Prilepko, D. G. Orlovskiy and I. A. Vasin, Methods for Solving Inverse Problems in Mathematical Physics, Marcel Dekker, New York, 2000. [17] A. G. Sveshnikov, A. B. Al'shin, M. O. Korpusov and Yu. D. Pletner, Lineinye i Nelineinye Uravneniya Sobolevskogo Tipa, (Russian) [Linear and Nonlinear Equations of the Sobolev Type], Fizmatlit, Moscow, 2007. [18] A. V. Urazaeva and V. E. Fedorov, An inverse problem for linear Sobolev type equations, J. Inverse Ill-Posed Probl., 12 (2004), 387-395. doi: 10.1515/1569394042248210. [19] A. V. Urazaeva and V. E. Fedorov, Prediction-control problem for some systems of equations of fluid dynamics, Differ. Equ., 44 (2008), 1147-1156. doi: 10.1134/S0012266108080120. [20] A. V. Urazaeva and V. E. Fedorov, On the well-posedness of the prediction-control problem for some systems of equations, Math. Notes, 85 (2009), 426-436. doi: 10.1134/S0001434609030134. [21] A. V. Urazaeva and V. E. Fedorov, Lineinaya evolutsionnaya obratnaya zadacha dlya uravnenii sobolevskogo tipa, (Russian) [Linear evolutionary inverse problem for Sobolev type equations], in Neklassicheskie uravnenia matematicheskoi fiziki (ed. A.I. Kozhanov), Sobolev Institute of Mathematics of the SB RAS, (2010), 293-310.
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