Geodesic
Solvability of nonlinear heat equation in class of unbounded functions with degeneration of coefficient near derivative with respect to time
Dalʹnevostočnyj matematičeskij žurnal, Tome 7 (2007) no. 1, pp. 3-16.

Voir la notice de l'article provenant de la source Math-Net.Ru

In this paper, we consider quasilinear parabolic equations which degenerate on a solution due to a multiplier of the derivative with respect to time. In the many-dimensional case, we prove the existence of a solution of a general boundary-value problem from a class of unbounded functions. Restrictions to nonlinearity of the multiplier of the derivative with respect to time are different from ones considered before by other authors. 
@article{DVMG_2007_7_1_a0,
     author = {E. G. Agapova},
     title = {Solvability of nonlinear heat equation in class of unbounded functions with degeneration of coefficient near derivative with respect to time},
     journal = {Dalʹnevosto\v{c}nyj matemati\v{c}eskij \v{z}urnal},
     pages = {3--16},
     publisher = {mathdoc},
     volume = {7},
     number = {1},
     year = {2007},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/DVMG_2007_7_1_a0/}
}
TY  - JOUR
AU  - E. G. Agapova
TI  - Solvability of nonlinear heat equation in class of unbounded functions with degeneration of coefficient near derivative with respect to time
JO  - Dalʹnevostočnyj matematičeskij žurnal
PY  - 2007
SP  - 3
EP  - 16
VL  - 7
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/DVMG_2007_7_1_a0/
LA  - ru
ID  - DVMG_2007_7_1_a0
ER  - 
%0 Journal Article
%A E. G. Agapova
%T Solvability of nonlinear heat equation in class of unbounded functions with degeneration of coefficient near derivative with respect to time
%J Dalʹnevostočnyj matematičeskij žurnal
%D 2007
%P 3-16
%V 7
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/DVMG_2007_7_1_a0/
%G ru
%F DVMG_2007_7_1_a0
E. G. Agapova. Solvability of nonlinear heat equation in class of unbounded functions with degeneration of coefficient near derivative with respect to time. Dalʹnevostočnyj matematičeskij žurnal, Tome 7 (2007) no. 1, pp. 3-16. http://geodesic.mathdoc.fr/item/DVMG_2007_7_1_a0/

[1] E. G. Agapova, A. G. Podgaev, “Issledovanie razreshimosti evolyutsionnogo vyrozhdayuschegosya uravneniya s neodnorodnoi nelineinostyu metodom kompaktnosti”, Differents. uravneniya, 35:6 (1999), 772–779 | MR | Zbl

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

[3] O. V. Gorevchuk, “Kraevaya zadacha dlya odnogo klassa nelineinykh vyrozhdayuschikhsya parabolicheskikh uravnenii”, Sib. mat. zhurn., 38:6 (1997), 1222–1239 | MR | Zbl

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

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

[6] G. I. Laptev, “Slabye resheniya kvazilineinykh parabolicheskikh uravnenii vtorogo poryadka s dvoinoi nelineinostyu”, Mat. sb., 188:9 (1997), 83–112 | MR | Zbl

[7] R. Suzuki, “Boundedness of global solutions of one dimensional quasilinear degenerate parabolic equations”, J. Math. Soc. Jap., 50:1 (1998), 119–138 | DOI | MR | Zbl

[8] A. G. Podgaev, “O granichnykh zadachakh dlya nekotorykh kvazilineinykh parabolicheskikh uravnenii s neklassicheskimi vyrozhdeniyami”, Sib. mat. zhurn., 28:2 (1987), 129–139 | MR | Zbl

[9] J. Carrillo, “Entropy solutions for nonlinear degenerate problems”, Arch. Rational Mech. Anal., 147 (1999), 269–361 | DOI | MR | Zbl

[10] A. V. Ivanov, P. Z. Mkrtchyan, V. Yaeger, “Suschestvovanie i edinstvennost regulyarnogo resheniya pervoi nachalno-kraevoi zadachi dlya nekotorogo klassa dvazhdy nelineinykh parabolicheskikh uravnenii”, Zap. nauch. seminarov POMI, 213, 1994, 48–65 | Zbl

[11] J.-B. Betbeder, “Etude d$^,$une equation non lineaire d$^,$evolution de type divergentiel”, Commun. Part. Differ. Equat., 19:7–8 (1994), 1019–1035 | DOI | MR | Zbl

[12] D. Eidus, “The Cauchy problem for the non-linear filtration equation in an inhomogeneous medium”, J. diff. eq., 84 (1990), 309–318 | DOI | MR | Zbl

[13] H. Kazuya, Y. Takaaki, “An ill-posed estimate of positive solutions of a degenerate nonlinear parabolic equation”, Tokyo J. Math., 19:2 (1996), 331–352 | DOI | MR | Zbl

[14] P. Urruty, “Sur une équation autonome d'évolution doublement non linéaire et dégénérée”, C. r. Acad. sci. ser. 1, 322:8 (1996), 741–744 | MR | Zbl

[15] S. N. Glazatov, “O nekotorykh zadachakh dlya dvazhdy nelineinykh parabolicheskikh uravnenii i uravnenii peremennogo tipa”, Matematicheskie trudy, 3:2 (2000), 71–110 | MR | Zbl

[16] O. A. Ladyzhenskaya, V. A. Solonnikova, N. N. Uraltseva, Lineinye i kvazilineinye uravneniya parabolicheskogo tipa, Nauka, M., 1967, 736 pp. | MR

[17] A. G. Podgaev, “Kompaktnost nekotorykh nelineinykh mnozhestv”, Dokl. AN SSSR, 285:5 (1985), 1064–1066 | MR | Zbl

[18] Y. Y. Li, M. Zhu, “Sharp Sobolev inequalities involving boundary terms”, Geometric and functional analysis, 8 (1998), 59–87 | DOI | MR | Zbl

[19] D. Gilbarg, N. Trudinger, Ellipticheskie differentsialnye uravneniya s chastnymi proizvodnymi vtorogo poryadka, Nauka, M., 1989, 464 pp. | MR | Zbl

[20] A. N. Kolmogorov, S. V. Fomin, Elementy teorii funktsii i funktsionalnogo analiza, M., 1981, 624 pp. | MR

[21] Zh.-L. Lions, Nekotorye metody resheniya nelineinykh kraevykh zadach, Mir, M., 1972, 588 pp. | MR