Geodesic
On a class of nonlocal parabolic equations
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 51 (2011) no. 6, pp. 1056-1063 Cet article a éte moissonné depuis la source Math-Net.Ru

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We consider a boundary value problem for parabolic equations with nonlocal nonlinearity of such a form that favorably differs from other equations in that it leads to partial differential equations that have important properties of ordinary differential equations. Local solvability and uniqueness theorems are proved, and an analog of the Painlevé singular nonfixed points theorem is proved. In this case, there is an alternative – either a solution exists for all $t\ge0$ or it goes to infinity in a finite time $t=T$ (blowup mode). Sufficient conditions for the existence of a blowup mode are given. 
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A. N. Bogolyubov; M. D. Malykh. On a class of nonlocal parabolic equations. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 51 (2011) no. 6, pp. 1056-1063. http://geodesic.mathdoc.fr/item/ZVMMF_2011_51_6_a7/

[1] Ladyzhenskaya O. A., Uraltseva N. N., “Kraevaya zadacha dlya lineinykh i kvazilineinykh parabolicheskikh uravnenii. I”, Izv. AN SSSR. Ser. matem., 26:1 (1962), 5–52 | MR

[2] Sveshnikov A. G., Alshin A. B., Korpusov M. O., Nelineinyi funktsionalnyi analiz i ego prilozheniya k uravneniyam v chastnykh proizvodnykh, Nauchn. mir, M., 2008

[3] Golubev V. V., Analiticheskaya teoriya differentsialnykh uravnenii, Fizmatiz, M.-L., 1950

[4] Gérard R., Sec A., “Fueiletages de Painlevé”, Bull. Soc. Math. France, 100:1 (1972), 47 | MR | Zbl

[5] Mitidieri E. L., Pokhozhaev S. I., “Apriornye otsenki i otsutstvie reshenii differentsialnykh neravenstv v chastnykh proizvodnykh”, Tr. MIRAN, 234, M., 2001

[6] Pokhozhaev S. I., “O razrushenii reshenii uravneniya Kuramoto–Sivashinskogo”, Matem. sb., 199:9 (2008), 97–106 | MR | Zbl