Recovery of the coefficient of $u_t$ in the heat equation from a condition of nonlocal observation in time

A. B. Kostin

Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 55 (2015) no. 1, pp. 89-104

Résumé

The inverse problem of finding the coefficient $\rho(x)=\rho_0+r(x)$ multiplying $u_t$ in the heat equation is studied. The unknown function $r(x)\geqslant0$ is sought in the class of bounded functions, and $\rho_0$ is a given positive constant. In addition to the initial and boundary conditions (data of the direct problem), a nonlocal observation condition is specified in the form $\int\limits_0^T u(x,t)d\mu(t)=\chi(x)$ with a given measure $d\mu(t)$ and a function $\chi(x)$. The case of integral observation (i.e., $d\mu(t)=\omega(t)dt$) is considered separately. Sufficient conditions for the existence and uniqueness of a solution to the inverse problem are obtained in the form of easy-to-check inequalities. Examples of inverse problems are given for which the assumptions of the theorems proved in this work are satisfied.

Bibliographie

[1] Cannon J. R., “Determination of an unknown coefficient in a parabolic differential equation”, Duke Math. J., 30:2 (1963), 313–323 | DOI

[2] Beznoschenko N. Ya., Prilepko A. I., “Obratnye zadachi dlya uravnenii parabolicheskogo tipa”, Problemy matematicheskoi fiziki i vychislitelnoi matematiki, Nauka, M., 1977, 51–63

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

[4] Prilepko A. I., Solovev V. V., “O razreshimosti obratnykh kraevykh zadach opredeleniya koeffitsienta pered mladshei proizvodnoi v parabolicheskom uravnenii”, Differents. ur-niya, 23:1 (1987), 136–143

[5] Isakov V. M., “Inverse parabolic problems with the final overdetermination”, Comm. on Pure and Appl. Math., 44 (1991), 185–209 | DOI

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

[7] Prilepko A. I., Kostin A. B., “Ob obratnykh zadachakh opredeleniya koeffitsienta v parabolicheskom uravnenii, II”, Sib. matem. zh., 34:5 (1993), 147–162

[8] Isakov V. M., Inverse problems for partial differential equations, Springer, New York, 2006

[9] Prilepko A. P., Tikhonov I. V., “Printsip pozitivnosti resheniya v lineinoi obratnoi zadache i ego primenenie k koeffitsientnoi zadache teploprovodnosti”, Dokl. AN, 364:1 (1999), 21–23

[10] Kozhanov A. I., “O razreshimosti obratnoi zadachi nakhozhdeniya koeffitsienta teploprovodnosti”, Sib. matem. zh., 46:5 (2005), 1053–1071

[11] Kamynin V. L., Kostin A. B., “Dve obratnye zadachi opredeleniya koeffitsienta v parabolicheskom uravnenii”, Differents. ur-niya, 46:3 (2010), 372–383

[12] Kamynin V. L., “Obratnaya zadacha opredeleniya koeffitsienta pered mladshei proizvodnoi v parabolicheskom uravnenii na ploskosti”, Differents. ur-niya, 48:2 (2012), 207–216

[13] Kamynin V. L., “Obratnaya zadacha opredeleniya mladshego koeffitsienta v parabolicheskom uravnenii pri uslovii integralnogo nablyudeniya”, Matem. zametki, 94:2 (2013), 207–217 | DOI

[14] Tikhonov A. N., Arsenin V. Ya., Metody resheniya nekorrektnykh zadach, Nauka, M., 1979

[15] Lavrentev M. M., Romanov V. G., Shishatskii S. P., Nekorrektnye zadachi matematicheskoi fiziki i analiza, Nauka, M., 1980

[16] Romanov V. G., Obratnye zadachi dlya differentsialnykh uravnenii, Izd-vo Novosibirsk. un-ta, Novosibirsk, 1973

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

[18] Prilepko A. I., Orlovsky D. G., Vasin I. A., Methods for solving inverse problems in mathematical physics, Marcel Dekker, New York, 2000

[19] Prilepko A. P., Tikhonov I. V., “Vosstanovlenie neodnorodnogo slagaemogo v abstraktnom evolyutsionnom uravnenii”, Izv. RAN. Ser. matem., 58:2 (1994), 167–188

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

[21] Ladyzhenskaya O. A., Kraevye zadachi matematicheskoi fiziki, Nauka, M., 1973

[22] Natanson I. P., Teoriya funktsii veschestvennoi peremennoi, Nauka, M., 1974

[23] Mikhailov V. P., Differentsialnye uravneniya v chastnykh proizvodnykh, Nauka, M., 1983

[24] Gilbarg D., Trudinger N., Ellipticheskie differentsialnye uravneniya s chastnymi proizvodnymi vtorogo poryadka, Nauka, M., 1989

[25] Kostin A. B., “Obratnaya zadacha vosstanovleniya istochnika v parabolicheskom uravnenii po usloviyu nelokalnogo nablyudeniya”, Matem. sb., 204:10 (2013), 3–46 | DOI

[26] Lieberman G. M., Second order parabolic differential equations, World Scientific, Singapore, 2005

[27] S. G. Krein (red.), Funktsionalnyi analiz, Nauka, M., 1964

[28] Lyusternik L. A., Sobolev V. I., Kratkii kurs funktsionalnogo analiza, Vysshaya shkola, M., 1982

[29] Birkgof G., Teoriya reshetok, Nauka, M., 1984