Geodesic
On global solvability of nonlinear parabolic boundary-value problems
Sbornik. Mathematics, Tome 26 (1975) no. 1, pp. 89-104 
A. V. Babin. On global solvability of nonlinear parabolic boundary-value problems. Sbornik. Mathematics, Tome 26 (1975) no. 1, pp. 89-104. http://geodesic.mathdoc.fr/item/SM_1975_26_1_a4/
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In this paper one considers nonlinear parabolic boundary-value problems of a general form. It is known that the solution of such problems can go to infinity in a finite interval of time. One shows that this effect is in a certain sense of a finite-dimensional character. Namely, one shows that if the solution is considered on the segment $[0,T]$, while the right-hand sides are bounded in the norm by a constant $R$ and satisfy a finite number of conditions, then the problem admits a solution which is smooth for $0\leqslant t\leqslant T$ (the number of conditions depends on $R$ and $T$). Bibliography: 11 titles.

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[2] A. V. Babin, “Konechnomernost yadra i koyadra kvazilineinykh ellipticheskikh otobrazhenii”, Matem. sb., 93 (135) (1974), 422–450 | MR | Zbl

[3] S. M. Nikolskii, Priblizhenie funktsii mnogikh peremennykh i teoremy vlozheniya, izd-vo «Nauka», Moskva, 1969 | MR

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[5] R. Pale, Seminar po teoreme Ati-Zingera ob indekse, izd-vo «Mir», Moskva, 1970 | MR

[6] O. A. Ladyzhenskaya, V. A. Solonnikov, N. N. Uraltseva, Lineinye i kvazilineinye uravneniya parabolicheskogo tipa, izd-vo «Nauka», Moskva, 1967

[7] O. A. Oleinik, S. N. Kruzhkov, “Kvazilineinye parabolicheskie uravneniya vtorogo poryadka so mnogimi nezavisimymi peremennymi”, Uspekhi matem. nauk, XVI:5 (101) (1961), 115–155 | MR

[8] A. F. Filippov, “Ob usloviyakh suschestvovaniya resheniya kvazilineinogo parabolicheskogo uravneniya”, DAN SSSR, 141:3 (1961), 568–570 | MR | Zbl

[9] V. A. Solonnikov, “Ob otsenkakh v $L^p$-reshenii ellipticheskikh i parabolicheskikh sistem”, Trudy Matem. in-ta im. V. A. Steklova, CII (1966), 137–160

[10] A. Friedman, “Mildly nonlinear parabolic equations with application to flow of gases through porous media”, Arch. Rat. Mech and Analysis, 5:3 (1960), 238–248 | DOI | MR | Zbl

[11] H. Fujita, “On some nonexistence and nonuniqueness theorems for nonlinear parabolic equations”, Proc. Symp. in pure math., part 1, XVIII, Amer. Math. Soc., 1970, 105–113 | MR