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 & 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. doi: 10.1515/jiip.2002.10.6.547. Google Scholar

[2]

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

[3]

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

[4]

G. V. Demidenko and S. V. Uspenskii, Partial Differential Equations and Systems not Solvable with Respect to the Highest-Order Derivative,, Marcel Dekker, (2003). doi: 10.1201/9780203911433. Google Scholar

[5]

A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces,, Marcel Dekker, (1999). Google Scholar

[6]

V. E. Fedorov, Linear equations of the Sobolev type with relatively $p$-radial operators,, Dokl. Akad. Nauk, 351 (1996), 316. Google Scholar

[7]

V. E. Fedorov, Degenerate strongly continuous semigroups of operators,, St. Petersburgh. Math. J., 12 (2001), 471. Google Scholar

[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. doi: 10.1007/s11202-005-0035-9. Google Scholar

[9]

V. E. Fedorov, Svoistva psevdoresolvent i usloviya sushchestvovaniya vyrozhdennoi polugruppy operatorov,, (Russian) [Pseudoresolvent properties and a degenerate operator semigroup existence conditions], 11 (2009), 12. Google Scholar

[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. Google Scholar

[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, 13 (2012), 50. Google Scholar

[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. Google Scholar

[13]

O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow,, New York-London-Paris: Gordon and Breach, (1969). Google Scholar

[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], 179 (1988), 126. Google Scholar

[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, (2013). Google Scholar

[16]

A. I. Prilepko, D. G. Orlovskiy and I. A. Vasin, Methods for Solving Inverse Problems in Mathematical Physics,, Marcel Dekker, (2000). Google Scholar

[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], (2007). Google Scholar

[18]

A. V. Urazaeva and V. E. Fedorov, An inverse problem for linear Sobolev type equations,, J. Inverse Ill-Posed Probl., 12 (2004), 387. doi: 10.1515/1569394042248210. Google Scholar

[19]

A. V. Urazaeva and V. E. Fedorov, Prediction-control problem for some systems of equations of fluid dynamics,, Differ. Equ., 44 (2008), 1147. doi: 10.1134/S0012266108080120. Google Scholar

[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. doi: 10.1134/S0001434609030134. Google Scholar

[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), (2010), 293. Google Scholar

show all references

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. doi: 10.1515/jiip.2002.10.6.547. Google Scholar

[2]

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

[3]

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

[4]

G. V. Demidenko and S. V. Uspenskii, Partial Differential Equations and Systems not Solvable with Respect to the Highest-Order Derivative,, Marcel Dekker, (2003). doi: 10.1201/9780203911433. Google Scholar

[5]

A. Favini and A. Yagi, Degenerate Differential Equations in Banach Spaces,, Marcel Dekker, (1999). Google Scholar

[6]

V. E. Fedorov, Linear equations of the Sobolev type with relatively $p$-radial operators,, Dokl. Akad. Nauk, 351 (1996), 316. Google Scholar

[7]

V. E. Fedorov, Degenerate strongly continuous semigroups of operators,, St. Petersburgh. Math. J., 12 (2001), 471. Google Scholar

[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. doi: 10.1007/s11202-005-0035-9. Google Scholar

[9]

V. E. Fedorov, Svoistva psevdoresolvent i usloviya sushchestvovaniya vyrozhdennoi polugruppy operatorov,, (Russian) [Pseudoresolvent properties and a degenerate operator semigroup existence conditions], 11 (2009), 12. Google Scholar

[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. Google Scholar

[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, 13 (2012), 50. Google Scholar

[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. Google Scholar

[13]

O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow,, New York-London-Paris: Gordon and Breach, (1969). Google Scholar

[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], 179 (1988), 126. Google Scholar

[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, (2013). Google Scholar

[16]

A. I. Prilepko, D. G. Orlovskiy and I. A. Vasin, Methods for Solving Inverse Problems in Mathematical Physics,, Marcel Dekker, (2000). Google Scholar

[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], (2007). Google Scholar

[18]

A. V. Urazaeva and V. E. Fedorov, An inverse problem for linear Sobolev type equations,, J. Inverse Ill-Posed Probl., 12 (2004), 387. doi: 10.1515/1569394042248210. Google Scholar

[19]

A. V. Urazaeva and V. E. Fedorov, Prediction-control problem for some systems of equations of fluid dynamics,, Differ. Equ., 44 (2008), 1147. doi: 10.1134/S0012266108080120. Google Scholar

[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. doi: 10.1134/S0001434609030134. Google Scholar

[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), (2010), 293. Google Scholar

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