Geodesic
On the Stabilization to Zero of the Solutions of the Inverse Problem for a Degenerate Parabolic Equation with Two Independent Variables
Matematičeskie zametki, Tome 101 (2017) no. 6, pp. 860-870.

Voir la notice de l'article provenant de la source Math-Net.Ru

Theorems on the stabilization to zero as $t\to +\infty$ of solutions of the inverse problem of determining the unknown right-hand side of a degenerate parabolic equation with one space variable belonging to a bounded closed interval are proved under an integral observation condition. Further, it is assumed that the coefficients of the equation are unbounded. These theorems can be used to obtain an upper bound for the stabilization rate. The paper contains examples of equations for which the assumptions of the theorems hold, as well as an example showing that such assumptions are essential.
Keywords: degenerate parabolic equation, asymptotic behavior, inverse problem. 
@article{MZM_2017_101_6_a6,
     author = {V. L. Kamynin},
     title = {On the {Stabilization} to {Zero} of the {Solutions} of the {Inverse} {Problem} for a {Degenerate} {Parabolic} {Equation} with {Two} {Independent} {Variables}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {860--870},
     publisher = {mathdoc},
     volume = {101},
     number = {6},
     year = {2017},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2017_101_6_a6/}
}
TY  - JOUR
AU  - V. L. Kamynin
TI  - On the Stabilization to Zero of the Solutions of the Inverse Problem for a Degenerate Parabolic Equation with Two Independent Variables
JO  - Matematičeskie zametki
PY  - 2017
SP  - 860
EP  - 870
VL  - 101
IS  - 6
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2017_101_6_a6/
LA  - ru
ID  - MZM_2017_101_6_a6
ER  - 
%0 Journal Article
%A V. L. Kamynin
%T On the Stabilization to Zero of the Solutions of the Inverse Problem for a Degenerate Parabolic Equation with Two Independent Variables
%J Matematičeskie zametki
%D 2017
%P 860-870
%V 101
%N 6
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2017_101_6_a6/
%G ru
%F MZM_2017_101_6_a6
V. L. Kamynin. On the Stabilization to Zero of the Solutions of the Inverse Problem for a Degenerate Parabolic Equation with Two Independent Variables. Matematičeskie zametki, Tome 101 (2017) no. 6, pp. 860-870. http://geodesic.mathdoc.fr/item/MZM_2017_101_6_a6/

[1] I. A. Vasin, V. L. Kamynin, “Ob asimptoticheskom povedenii reshenii obratnykh zadach dlya parabolicheskikh uravnenii”, Sib. matem. zhurn., 38:4 (1997), 750–766 | MR | Zbl

[2] I. A. Vasin, V. L. Kamynin, “Asimptoticheskoe povedenie reshenii obratnykh zadach dlya parabolicheskikh uravnenii s neregulyarnymi koeffitsientami”, Matem. sb., 188:3 (1997), 49–64 | DOI | MR | Zbl

[3] R. Riganti, E. Savateev, “Solution of an inverse problem for the nonlinear heat equation”, Comm. Partial Differential Equations, 19:9-10 (1994), 1611–1628 | DOI | MR | Zbl

[4] A. F. Güvenilir, V. K. Kalantarov, “The asymptotic behavior of solutions to an inverse problem for differential operator equations”, Math. Comput. Modelling, 37:9-10 (2003), 907–914 | DOI | MR | Zbl

[5] Ö. F. Gözükızıl, M. Yaman, “Long-time behaviour of the solutions to inverse problems for parabolic equations”, Appl. Math. Comput., 184:2 (2007), 669–673 | DOI | MR | Zbl

[6] D. Guidetti, “Asymptotic expansion of solutions to an inverse problem of parabolic type with nonhomogeneous boundary conditions”, Proc. Roy. Soc. Edinburgh Sect. A, 141:4 (2011), 777–817 | DOI | MR | Zbl

[7] D. Guidetti, “Convergence to a stationary state of solutions to inverse problems of parabolic type”, Discrete Contin. Dyn. Syst. Ser. S, 6:3 (2013), 711–722 | DOI | MR | Zbl

[8] V. L. Kamynin, “O korrektnoi razreshimosti obratnoi zadachi dlya vyrozhdayuschikhsya parabolicheskikh uravnenii pri uslovii integralnogo nablyudeniya”, Tezisy dokladov mezhdunarodnogo nauchnogo seminara po obratnym i nekorrektno postavlennym zadacham (Moskva, 19–21 noyabrya 2015 g.), RUDN, M., 2015, 90–91

[9] V. L. Kamynin, “Obratnaya zadacha opredeleniya pravoi chasti v vyrozhdayuschemsya parabolicheskom uravnenii s neogranichennymi koeffitsientami”, Zh. vychisl. matem. i matem. fiz., 57:5 (2017), 832–841

[10] V. L. Kamynin, “O korrektnoi razreshimosti obratnoi zadachi opredeleniya pravoi chasti v vyrozhdayuschemsya parabolicheskom uravnenii s usloviem integralnogo nablyudeniya”, Matem. zametki, 98:5 (2015), 710–724 | DOI | MR | Zbl

[11] O. A. Ladyzhenskaya, V. A. Solonnikov, N. N. Uraltseva, Lineinye i kvazilineinye uravneniya parabolicheskogo tipa, Nauka, M., 1967 | MR | Zbl

[12] A. M. Ilin, A. S. Kalashnikov, O. A. Oleinik, “Lineinye uravneniya vtorogo poryadka parabolicheskogo tipa”, UMN, 17:3 (105) (1962), 3–146 | MR | Zbl

[13] A. Fridman, Uravneniya s chastnymi proizvodnymi parabolicheskogo tipa, M., Nauka, 1968 | MR | Zbl