Geodesic
On the Inverse Problem of Determining the Leading Coefficient in Parabolic Equations
Matematičeskie zametki, Tome 84 (2008) no. 1, pp. 48-58 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We study the unique solvability of the inverse problem of determining the leading coefficient in the parabolic equation on the plane with coefficients depending on both time and spatial variables under the condition of integral overdetermination with respect to time. We obtain sufficient conditions for the unique solvability of the inverse problem. We present nontrivial examples of problems for which such conditions hold. It is shown that the imposed conditions necessarily hold if either the time interval is sufficiently large or the space interval on which the problem is considered is sufficiently small.
Mots-clés : parabolic equation
Keywords: inverse problem for the parabolic equation, Poincaré–Steklov inequality, Schauder fixed-point theorem, maximum principle, compact operator. 
@article{MZM_2008_84_1_a3,
     author = {V. L. Kamynin},
     title = {On the {Inverse} {Problem} of {Determining} the {Leading} {Coefficient} in {Parabolic} {Equations}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {48--58},
     year = {2008},
     volume = {84},
     number = {1},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2008_84_1_a3/}
}
TY  - JOUR
AU  - V. L. Kamynin
TI  - On the Inverse Problem of Determining the Leading Coefficient in Parabolic Equations
JO  - Matematičeskie zametki
PY  - 2008
SP  - 48
EP  - 58
VL  - 84
IS  - 1
UR  - http://geodesic.mathdoc.fr/item/MZM_2008_84_1_a3/
LA  - ru
ID  - MZM_2008_84_1_a3
ER  - 
%0 Journal Article
%A V. L. Kamynin
%T On the Inverse Problem of Determining the Leading Coefficient in Parabolic Equations
%J Matematičeskie zametki
%D 2008
%P 48-58
%V 84
%N 1
%U http://geodesic.mathdoc.fr/item/MZM_2008_84_1_a3/
%G ru
%F MZM_2008_84_1_a3
V. L. Kamynin. On the Inverse Problem of Determining the Leading Coefficient in Parabolic Equations. Matematičeskie zametki, Tome 84 (2008) no. 1, pp. 48-58. http://geodesic.mathdoc.fr/item/MZM_2008_84_1_a3/

[1] A. I. Prilepko, A. B. Kostin, “Ob obratnykh zadachakh opredeleniya koeffitsienta v parabolicheskom uravnenii. I”, Sib. matem. zhurn., 33:3 (1992), 146–155 | MR | Zbl

[2] A. I. Kozhanov, “O razreshimosti obratnoi zadachi nakhozhdeniya koeffitsienta teploprovodnosti”, Sib. matem. zhurn., 46:5 (2005), 1053–1071 | MR | Zbl

[3] V. M. Isakov, “Ob odnom klasse obratnykh zadach dlya parabolicheskikh uravnenii”, Dokl. AN SSSR, 263:6 (1982), 1296–1299 | MR | Zbl

[4] V. Isakov, “Inverse parabolic problems with the final overdetermination”, Comm. Pure Appl. Math., 44:2 (1991), 185–209 | DOI | MR | Zbl

[5] A. M. Denisov, Vvedenie v teoriyu obratnykh zadach, Izd-vo MGU, M., 1994

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

[7] S. N. Kruzhkov, “Kvazilineinye parabolicheskie uravneniya i sistemy s dvumya nezavisimymi peremennymi”, Tr. sem. im. I. G. Petrovskogo, 1979, no. 5, 217–272 | MR | Zbl

[8] S. N. Kruzhkov, Nelineinye uravneniya s chastnymi proizvodnymi, ch. 1: Ellipticheskie i parabolicheskie uravneniya vtorogo poryadka, Izd-vo MGU, M., 1969

[9] L. A. Lyusternik, V. I. Sobolev, Kratkii kurs funktsionalnogo analiza, Vysshaya shkola, M., 1982 | MR | Zbl