Geodesic
Weak solutions of second-order quasilinear parabolic equations with double non-linearity
Sbornik. Mathematics, Tome 188 (1997) no. 9, pp. 1343-1370 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The first boundary-value problem for the equation $$ \beta (u)\frac {\partial u}{\partial t}-\sum _{i=1}^nD_iA_i(t,x,u,Du)+ A_0(t,x,u,Du)=0 $$ is considered in a bounded subdomain of $n$. The function $\beta (u)$ is assumed to be continuous and satisfy the following growth conditions: $$ c|u|^{r-2}\leqslant \beta (u)\leqslant C\bigl (|u|^{r-2}+1\bigr ),\qquad r\geqslant 2. $$ The other coefficients satisfy the standard conditions of the theory of monotone operators. An existence theorem for a global weak solution is proved. 
@article{SM_1997_188_9_a4,
     author = {G. I. Laptev},
     title = {Weak solutions of second-order quasilinear parabolic equations with double non-linearity},
     journal = {Sbornik. Mathematics},
     pages = {1343--1370},
     year = {1997},
     volume = {188},
     number = {9},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1997_188_9_a4/}
}
TY  - JOUR
AU  - G. I. Laptev
TI  - Weak solutions of second-order quasilinear parabolic equations with double non-linearity
JO  - Sbornik. Mathematics
PY  - 1997
SP  - 1343
EP  - 1370
VL  - 188
IS  - 9
UR  - http://geodesic.mathdoc.fr/item/SM_1997_188_9_a4/
LA  - en
ID  - SM_1997_188_9_a4
ER  - 
%0 Journal Article
%A G. I. Laptev
%T Weak solutions of second-order quasilinear parabolic equations with double non-linearity
%J Sbornik. Mathematics
%D 1997
%P 1343-1370
%V 188
%N 9
%U http://geodesic.mathdoc.fr/item/SM_1997_188_9_a4/
%G en
%F SM_1997_188_9_a4
G. I. Laptev. Weak solutions of second-order quasilinear parabolic equations with double non-linearity. Sbornik. Mathematics, Tome 188 (1997) no. 9, pp. 1343-1370. http://geodesic.mathdoc.fr/item/SM_1997_188_9_a4/

[1] Lions Zh.-L., Nekotorye metody resheniya nelineinykh kraevykh zadach, Mir, M., 1972 | MR

[2] Raviart P. A., “Sur la résolution de certaines équations paraboliques non linéaires”, J. Funct. Anal., 5:2 (1970), 299–328 | DOI | MR | Zbl

[3] Ivanov A. V., “Kvazilineinye parabolicheskie uravneniya, dopuskayuschie dvoinoe vyrozhdenie”, Algebra i analiz, 4:6 (1992), 114–130 | MR | Zbl

[4] Kalashnikov A. S., “Nekotorye voprosy kachestvennoi teorii nelineinykh vyrozhdayuschikhsya parabolicheskikh uravnenii vtorogo poryadka”, UMN, 42:2 (1987), 135–176 | MR | Zbl

[5] Alt H. W., Luckhaus S., “Quasilinear elliptic-parabolic differential equations”, Math. Z., 183:3 (1983), 311–341 | DOI | MR | Zbl

[6] Pavlova M. F., “Issledovanie uravnenii nestatsionarnoi nelineinoi filtratsii”, Differents. uravneniya, 23:8 (1987), 1436–1446 | MR

[7] Leonov K. Ya., O nachalno-kraevoi zadache dlya odnogo klassa kvazilineinykh parabolicheskikh uravnenii, Preprint Instituta fiziki AN AzSSR No. 202, Institut fiziki AN AzSSR, Baku, 1987, s. 1–44 | MR

[8] Bernis F., “Existence results for doubly nonlinear higher order parabolic equations on unbounded domains”, Math. Ann., 279 (1988), 373–394 | DOI | MR | Zbl

[9] Blanchard D., Francfort G. A., “A few results on a class of degenerate parabolic equations”, Ann. Sc. norm. super. Pisa. Cl. sci., 18:2 (1991), 213–219 | MR

[10] Blanchard D., Redwane H., “Solutions renormalisées d'équations paraboliques à deux non linéarités”, C. R. Acad. Sci. Ser. 1, 319:8 (1994), 831–835 | MR | Zbl

[11] Soltanov K. N., “Suschestvovanie i nesuschestvovanie globalnogo resheniya nekotorykh nelineinykh elliptiko-parabolicheskikh uravnenii”, Differents. uravneniya, 29:4 (1993), 646–661 | MR | Zbl

[12] Ivanov A. V., Mkrtychyan P. Z., “O suschestvovanii nepreryvnykh po Gelderu obobschennykh reshenii pervoi kraevoi zadachi dlya kvazilineinykh parabolicheskikh uravnenii, dopuskayuschikh dvoinoe vyrozhdenie”, Zapiski nauch. sem. POMI, 183, POMI, SPb., 1990, 5–28 | MR

[13] Ivanov A. V., Mkrtychyan P. Z., Yaeger V., “Suschestvovanie i edinstvennost regulyarnogo resheniya pervoi nachalno-kraevoi zadachi dlya nekotorogo klassa dvazhdy nelineinykh parabolicheskikh uravnenii”, Zapiski nauch. sem. POMI, 213, POMI, SPb., 1994, 48–65 | MR

[14] Gaevskii Kh., Grëger K., Zakharias K., Nelineinye operatornye uravneniya i operatornye differentsialnye uravneniya, Nauka, M., 1978

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

[16] Danford N., Shvarts Dzh. T., Lineinye operatory. Obschaya teoriya, IL, M., 1962

[17] Nikolskii S. M., Priblizhenie funktsii mnogikh peremennykh i teoremy vlozheniya, Nauka, M., 1977 | MR

[18] Vainberg M. M., Variatsionnyi metod i metod monotonnykh operatorov, Nauka, M., 1972 | MR | Zbl