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Mat. Zametki, 2004, Volume 76, Issue 6, Pages 840–853 (Mi mz156)  

This article is cited in 34 scientific papers (total in 34 papers)

A Nonlinear Loaded Parabolic Equation and a Related Inverse Problem

A. I. Kozhanov

Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences

Abstract: The solvability of the nonlocal-in-time boundary-value problem for the nonlinear parabolic equation $u_t-\Delta u+c(\bar u(x,T))u=f(x,t)$, where $\bar u(x,t)= \alpha(t)u(x,t)+\int^t_0\beta(\tau)u(x,\tau) d\tau$ for given functions $\alpha(t)$ and $\beta(t)$, is studied. Existence and uniqueness theorems for regular solutions are proved; it is shown that the results obtained can be used to study the solvability of coefficient inverse problems.

DOI: https://doi.org/10.4213/mzm156

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English version:
Mathematical Notes, 2004, 76:6, 784–795

Bibliographic databases:

UDC: 517.946
Received: 26.12.2001

Citation: A. I. Kozhanov, “A Nonlinear Loaded Parabolic Equation and a Related Inverse Problem”, Mat. Zametki, 76:6 (2004), 840–853; Math. Notes, 76:6 (2004), 784–795

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. M. T. Dzhenaliev, M. I. Ramazanov, “On the boundary value problem for the spectrally loaded heat conduction operator”, Siberian Math. J., 47:3 (2006), 433–451  mathnet  crossref  mathscinet  zmath  isi
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    3. A. I. Kozhanov, “O razreshimosti obratnykh zadach vosstanovleniya koeffitsientov v uravneniyakh sostavnogo tipa”, Vestn. NGU. Ser. matem., mekh., inform., 8:3 (2008), 81–99  mathnet
    4. S. G. Pyatkov, B. N. Tsybikov, “Some classes of inverse evolution problems for parabolic equations”, Siberian Math. J., 50:1 (2009), 141–153  mathnet  crossref  mathscinet  isi
    5. A. I. Kozhanov, “On the Solvability of Boundary Value Problems for the Quasilinear Ultraparabolic Equations of Certain Mathematical Models of the Dynamics of Biological Systems”, J. Appl. Industr. Math., 4:4 (2010), 512–525  mathnet  crossref  mathscinet
    6. D. M. Akhmanova, M. T. Dzhenaliev, M. I. Ramazanov, “On a particular second kind Volterra integral equation with a spectral parameter”, Siberian Math. J., 52:1 (2011), 1–10  mathnet  crossref  mathscinet  isi
    7. Frost Marina B., “Razreshimost nachalno-kraevoi zadachi dlya uravneniya, opisyvayuschego volnu nagruzheniya”, Vestnik sibirskogo gosudarstvennogo aerokosmicheskogo universiteta im. akademika M.F. Reshetneva, 2011, no. 2, 80–81  elib
    8. Amangalieva M.M., Akhmanova D.M., Dzhenaliev M.T., Ramazanov M.I., “Boundary value problems for a spectrally loaded heat operator with load line approaching the time axis at zero or infinity”, Differ Equ, 47:2 (2011), 231–243  crossref  mathscinet  zmath  isi  elib  scopus
    9. Telesheva L.A., “O razreshimosti obratnoi zadachi dlya parabolicheskogo uravneniya vysokogo poryadka s neizvestnym koeffitsientom pri proizvodnoi po vremeni”, Matematicheskie zametki YaGU, 18:2 (2011), 180–201  zmath  elib
    10. Kosheleva Yu.A., “O razreshimosti nekotorykh lineinykh obratnykh zadach dlya ultraparabolicheskikh uravnenii”, Matematicheskie zametki YaGU, 18:2 (2011), 79–98  mathscinet  zmath  elib
    11. Nikolaev N.N., “Issledovanie obratnoi zadachi dlya uravneniya tretego poryadka”, Matematicheskie zametki YaGU, 19:1 (2012), 70–81  zmath  elib
    12. V. L. Kamynin, “The Inverse Problem of Determining the Lower-Order Coefficient in Parabolic Equations with Integral Observation”, Math. Notes, 94:2 (2013), 205–213  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
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    14. Lyubanova A.Sh., “On Nonlocal Problems For Systems of Parabolic Equations”, J. Math. Anal. Appl., 421:2 (2015), 1767–1778  crossref  mathscinet  zmath  isi  scopus  scopus
    15. A. Sh. Lyubanova, “On some boundary value problems for systems of pseudoparabolic equations”, Siberian Math. J., 56:4 (2015), 662–677  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    16. M. M. Amangalieva, M. T. Dzhenaliev, M. T. Kosmakova, M. I. Ramazanov, “On one homogeneous problem for the heat equation in an infinite angular domain”, Siberian Math. J., 56:6 (2015), 982–995  mathnet  crossref  crossref  mathscinet  isi  elib
    17. N. S. Popov, “O razreshimosti prostranstvenno nelokalnykh kraevykh zadach dlya odnomernykh psevdoparabolicheskikh i psevdogiperbolicheskikh uravnenii”, Vestn. SamGU. Estestvennonauchn. ser., 2015, no. 3(125), 29–43  mathnet  elib
    18. Korzyuk V.I., Dzhenaliev M.T., Kozlovskaya I.S., “Boundary Value Problems For a Weakly Loaded Operator of a Second-Order Hyperbolic Equation in a Cylindrical Domain”, Differ. Equ., 51:12 (2015), 1595–1606  crossref  mathscinet  zmath  isi  scopus  scopus
    19. Moiseev E.I., Yurchuk N.I., “Classical and Generalized Solutions of Problems For the Telegraph Equation With a Dirac Potential”, Differ. Equ., 51:10 (2015), 1330–1337  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    20. Baranovskaya S.N., Yurchuk N.I., “Cauchy Problem and the Second Mixed Problem For Parabolic Equations With the Dirac Potential”, Differ. Equ., 51:6 (2015), 819–821  crossref  mathscinet  zmath  isi  scopus  scopus
    21. Amangaliyeva M.M., Jenaliyev M.T., Kosmakova M.T., Ramazanov M.I., “Uniqueness and Non-Uniqueness of Solutions of the Boundary Value Problems of the Heat Equation”, Advancements in Mathematical Sciences (Ams 2015), AIP Conference Proceedings, 1676, eds. Ashyralyev A., Malkowsky E., Lukashov A., Basar F., Amer Inst Physics, 2015, 020028  crossref  isi  scopus
    22. A. I. Kozhanov, “Obratnye zadachi vosstanovleniya pravoi chasti spetsialnogo vida v parabolicheskom uravnenii”, Matematicheskie zametki SVFU, 23:4 (2016), 31–45  mathnet  elib
    23. Yu. A. Kosheleva, “O razreshimosti nekotorykh nelineinykh obratnykh zadach dlya ultraparabolicheskikh uravnenii”, Sib. elektron. matem. izv., 13 (2016), 1187–1206  mathnet  crossref
    24. A. I. Dreglya, N. A. Sidorov, “Identifikatsiya dinamiki vneshnei sily pri modelirovanii kolebanii”, Izvestiya Irkutskogo gosudarstvennogo universiteta. Seriya Matematika, 19 (2017), 105–112  mathnet  crossref
    25. S. Z. Dzhamalov, “Lineinaya obratnaya zadacha dlya uravneniya smeshannogo tipa vtorogo roda vtorogo poryadka s nelokalnymi granichnymi usloviyami v trekhmernom prostranstve”, Vestnik KRAUNTs. Fiz.-mat. nauki, 2017, no. 1(17), 7–13  mathnet  crossref  mathscinet  elib
    26. Khankishiyev Z., “On Solution of a Nonlocal Problem With Dynamic Boundary Conditions For a Loaded Linear Parabolic Equation By Straight-Line Methods”, Bull. Comput. Appl. Math., 5:1 (2017), 77–98  mathscinet  isi
    27. Nikolaev N.N., Popov S.V., “On the Problem of Recovering the Coefficients in a One-Dimensional Third Order Equation”, Proceedings of the 8th International Conference on Mathematical Modeling (ICMM-2017), AIP Conference Proceedings, 1907, eds. Egorov I., Popov S., Vabishchevich P., Antonov M., Lazarev N., Troeva M., Troeva M., Ivanova A., Gri, Amer Inst Physics, 2017, UNSP 030005  crossref  isi  scopus  scopus
    28. Cao K., Lesnic D., “Reconstruction of the Space-Dependent Perfusion Coefficient From Final Time Or Time-Average Temperature Measurements”, J. Comput. Appl. Math., 337 (2018), 150–165  crossref  mathscinet  zmath  isi  scopus  scopus
    29. Baranovskaya S.N., Novikov E.N., Yurchuk N.I., “Directional Derivative Problem For the Telegraph Equation With a Dirac Potential”, Differ. Equ., 54:9 (2018), 1147–1155  crossref  isi  scopus
    30. Aliona I. Dreglea, Nikolay A. Sidorov, “Integral equations in identification of external force and heat source density dynamics”, Bul. Acad. Ştiinţe Repub. Mold. Mat., 2018, no. 3, 68–77  mathnet
    31. S. N. Sidorov, “Obratnye zadachi dlya uravneniya smeshannogo parabolo-giperbolicheskogo tipa s vyrozhdayuscheisya parabolicheskoi chastyu”, Sib. elektron. matem. izv., 16 (2019), 144–157  mathnet  crossref
    32. Cao K., Lesnic D., “Simultaneous Reconstruction of the Perfusion Coefficient and Initial Temperature From Time-Average Integral Temperature Measurements”, Appl. Math. Model., 68 (2019), 523–539  crossref  mathscinet  isi  scopus
    33. S. N. Sidorov, “Inverse problems for a degenerate mixed parabolic-hyperbolic equation on finding time-depending factors in right hand sides”, Ufa Math. J., 11:1 (2019), 75–89  mathnet  crossref  isi
    34. Baranovskaya S.N., Yurchuk N.I., “Cauchy Problem and the Second Mixed Problem For Parabolic Equations With a Dirac Potential Concentrated At Finitely Many Given Points”, Differ. Equ., 55:3 (2019), 348–352  crossref  isi  scopus
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