Geodesic
Exponential solubility classes in a problem for the heat equation with a non-local condition for the time averages
Sbornik. Mathematics, Tome 196 (2005) no. 9, pp. 1319-1348 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

A non-local problem (with respect to time) for the heat equation is considered for $x\in\mathbb R^n$, $ 0\leqslant t\leqslant T$: find a function $u(x,t)$ such that $$ \frac{\partial u}{\partial t}=\Delta u,\qquad \frac1T\int_0^Tu(x,t)\,dt=\varphi(x). $$ An explicit formula for the solution is found. The question of its applicability is discussed. A description of well-posedness classes is presented. The main conjecture is as follows: as $|x|\to\infty$, the solution $u(x,t)$ grows no more rapidly than $\exp(\sigma|x|)$ with $\sigma<\sqrt{\pi/T}$ . 
@article{SM_2005_196_9_a3,
     author = {A. Yu. Popov and I. V. Tikhonov},
     title = {Exponential solubility classes in a problem for the~heat equation with a~non-local condition for the~time averages},
     journal = {Sbornik. Mathematics},
     pages = {1319--1348},
     year = {2005},
     volume = {196},
     number = {9},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2005_196_9_a3/}
}
TY  - JOUR
AU  - A. Yu. Popov
AU  - I. V. Tikhonov
TI  - Exponential solubility classes in a problem for the heat equation with a non-local condition for the time averages
JO  - Sbornik. Mathematics
PY  - 2005
SP  - 1319
EP  - 1348
VL  - 196
IS  - 9
UR  - http://geodesic.mathdoc.fr/item/SM_2005_196_9_a3/
LA  - en
ID  - SM_2005_196_9_a3
ER  - 
%0 Journal Article
%A A. Yu. Popov
%A I. V. Tikhonov
%T Exponential solubility classes in a problem for the heat equation with a non-local condition for the time averages
%J Sbornik. Mathematics
%D 2005
%P 1319-1348
%V 196
%N 9
%U http://geodesic.mathdoc.fr/item/SM_2005_196_9_a3/
%G en
%F SM_2005_196_9_a3
A. Yu. Popov; I. V. Tikhonov. Exponential solubility classes in a problem for the heat equation with a non-local condition for the time averages. Sbornik. Mathematics, Tome 196 (2005) no. 9, pp. 1319-1348. http://geodesic.mathdoc.fr/item/SM_2005_196_9_a3/

[1] Holmgren E., “Sur les solutions quasianalytiques de l'équation de la chaleur”, Ark. Mat., 18:9 (1924), 1–9

[2] Tychonoff A., “Théorèmes d'unicité pour l'équation de la chaleur”, Matem. sb., 42:2 (1935), 199–216 | Zbl

[3] Täcklind S., “Sur les classes quasianalytiques des solutions des équations aux dérivées partielles du type parabolique”, Nova Acta Reg. Soc. Sci. Upsaliensis. Ser. IV, 10:3 (1936), 1–57 | Zbl

[4] Gelfand I. M., Shilov G. E., Obobschennye funktsii. Vyp 3. Nekotorye voprosy teorii differentsialnykh uravnenii, GIFML, M., 1958

[5] Tikhonov I. V., “Teoremy edinstvennosti v lineinykh nelokalnykh zadachakh dlya abstraktnykh differentsialnykh uravnenii”, Izv. RAN. Ser. matem., 67:2 (2003), 133–166 | MR | Zbl

[6] Popov A. Yu., Tikhonov I. V., “Eksponentsialnye klassy edinstvennosti v zadachakh teploprovodnosti”, Dokl. RAN, 389:4 (2003), 465–467 | MR | Zbl

[7] Popov A. Yu., Tikhonov I. V., “Klassy edinstvennosti v nelokalnoi po vremeni zadache dlya uravneniya teploprovodnosti i kompleksnye sobstvennye funktsii operatora Laplasa”, Differents. uravneniya, 40:3 (2004), 396–405 | MR | Zbl

[8] Volevich L. R., Gindikin S. G., Obobschennye funktsii i uravneniya v svertkakh, Fizmatlit, M., 1994 | MR | Zbl

[9] Mikhailov V. P., Differentsialnye uravneniya v chastnykh proizvodnykh, Nauka, M., 1983 | MR

[10] Privalov I. I., Subgarmonicheskie funktsii, GRTTL, M., 1937

[11] Gyunter N. M., Teoriya potentsiala i ee primenenie k osnovnym zadacham matematicheskoi fiziki, GITTL, M., 1953 | MR

[12] Beitmen G., Erdeii A., Vysshie transtsendentnye funktsii. T. 2. Funktsii Besselya, funktsii parabolicheskogo tsilindra, ortogonalnye mnogochleny, Nauka, M., 1974

[13] Abramovits M., Stigan I. (red.), Spravochnik po spetsialnym funktsiyam s formulami, grafikami i matematicheskimi tablitsami, Nauka, M., 1979 | MR

[14] Prudnikov A. P., Brychkov Yu. A., Marichev O. I., Integraly i ryady. Dopolnitelnye glavy, Nauka, M., 1986 | MR

[15] Matematicheskaya entsiklopediya, 1. A–G, SE, M., 1977

[16] Vatson G. N., Teoriya besselevykh funktsii. Chast pervaya, IL, M., 1949

[17] Fedoryuk M. V., Asimptotika: Integraly i ryady, Nauka, M., 1987 | MR