Existence and uniqueness of solutions of degenerate parabolic equations with variable exponents of nonlinearity

S. N. Antontsev ; S. I. Shmarev

Fundamentalʹnaâ i prikladnaâ matematika, Tome 12 (2006) no. 4, pp. 3-19.

Résumé

We prove the existence and uniqueness of weak solutions of the Dirichlet problem for the nonlinear degenerate parabolic equations $$ u_{t}=\operatorname{div}(a|u|^{\gamma(x,t)}\nabla u)+\mathbf{b}|u|^{\gamma(x,t)/2}\nabla u-c|u|^{\sigma (x,t)-2}u+d, $$ where $a$, $\mathbf{b}$, $c$, and $d$ are given functions of the arguments $x$, $t$, and $u(x,t)$, and the exponents of nonlinearity $\gamma(x,t)$ and $\sigma(x,t)$ are known measurable and bounded functions of their arguments.

Bibliographie

[1] Antontsev S. N., Lokalizatsiya reshenii vyrozhdayuschikhsya uravnenii mekhaniki sploshnoi sredy, AN SSSR, Sibirskoe Otdelenie, Novosibirsk, 1986 | MR

[2] Antontsev S. N., Dias Kh. I., Domanskii A. V., “Ustoichivost i stabilizatsiya obobschënnykh reshenii vyrozhdayuschikhsya zadach dvukhfaznoi filtratsii”, Dokl. RAN, 325 (1992), 1151–1155 | MR | Zbl

[3] Antontsev S. N., “Nepreryvnaya zavisimost i stabilizatsiya reshenii vyrozhdayuscheisya sistemy uravnenii dvukhfaznoi filtratsii.”, Dinamika sploshnoi sredy. Dinamika zhidkosti so svobodnymi granitsami, 107, 1993, 11–25 | MR | Zbl

[4] Antontsev S. N., Domanskii A. V., “Edinstvennost obobschënnykh reshenii vyrozhdennoi zadachi dvukhfaznoi filtratsii”, Chislennye metody mekhaniki sploshnoi sredy, 15, no. 6, 1984, 15–27 | MR | Zbl

[5] Antontsev S. N., Domanskii A. V., Penkovskii V. I., Filtratsiya v priskvazhinnoi zone plasta i problemy intensifikatsii pritoka., AN SSSR, Sibirskoe Otdelenie, Institut gidrodinamiki, Novosibirsk, 1989 | Zbl

[6] Antontsev S. N., Kazhikhov A. V., Monakhov V. N., Kraevye zadachi mekhaniki neodnorodnoi zhidkosti, Nauka, Novosibirsk, 1983 | Zbl

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

[8] Kalashnikov A. S., “Nelineinye effekty pri rasprostranenii tepla v srede s istochnikami i stokami, blizkie lineinym”, Differents. uravn., 31:2 (1995), 277–288 | MR | Zbl

[9] Kalashnikov A. S., “O nekotorykh zadachakh nelineinoi teorii teploprovodnosti s dannymi, soderzhaschimi malyi parametr v eksponentakh”, ZhVM i MF, 35:7 (1995), 1077–1094 | MR | Zbl

[10] Kalashnikov A. S., “O nekotorykh nelineinykh zadachakh matematicheskoi fiziki s eksponentami, blizkimi k kriticheskim”, Tr. seminara im. I. G. Petrovskogo, no. 19, 1996, 73–98 | MR | Zbl

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

[12] Ladyzhenskaya O. A., Uraltseva N. N., Lineinye i kvazilineinye uravneniya ellipticheskogo tipa, Nauka, M., 1973 | MR

[13] Acerbi A., Mignione G., Seregin G., “Regularity results for a class of parabolic systems related to a class of non-newtonian fluids”, Ann. Inst. H. Poincaré, 21:6 (2004), 25–60 | MR | Zbl

[14] Antontsev S. N., Díaz J. I., Shmarev S., “Energy Methods for Free Boundary Problems”, Applications to Non-Linear PDEs and Fluid Mechanics, Bikhäuser, Boston, 2002 | MR

[15] Antontsev S. N., Gagneux G., “Petits paramètres et passages à la limite dans les problèmes de filtration diphasique”, Progress in Partial Differential Equations (Papers from the 3rd European Conf. on Elliptic and Parabolic Problems, Pont-à-Mousson, France, June, 1997), Pitman Res. Notes Math. Ser., 383, ed. H. Amann, Longman, Harlow, 1998, 18–27 | MR | Zbl

[16] Antontsev S. N., Shmarev S., “A model porous medium equation with variable exponents of nonlinearity: Existence, uniqueness and localization properties of solutions”, Nonlinear Anal., Theory Methods Appl., 60:3 (A) (2005), 515–545 | DOI | MR | Zbl

[17] Antontsev S. N., Shmarev S., “A generalized porous medium equation with variable exponents of nonlinearity: Existence, uniqueness and localization properties of solutions”, Int. Conf. “Differential Equations and Related Topics” dedicated to I. G. Petrovskii (2004), Book of Abstracts, 14–15

[18] Aronson D. G., “The porous medium equation”, Nonlinear Diffusion Problems (Lect. 2nd, 1985 Sess. C.I.M.E., Montecatini Terme/Italy), Lect. Notes Math., 1224, Springer, Berlin, 1985, 1–46 | MR

[19] Bénilan P., Boccardo L., Herrero M. A., “On the limit of solutions of $u_t=\Delta u^m$ as $m\to\infty$”, Rend. Sem. Mat. Univ. Politec. Torino, 1991, 1–13 | MR

[20] Chasseigne E., Vazquez J. L., “Theory of extended solutions for fast-diffusion equations in optimal classes of data. Radiation from Singularities”, Arch. Ration. Mech. Anal., 164:2 (2002), 133–187 | DOI | MR | Zbl

[21] Elliott C. M., Herrero M. A., King J. R., Ockendon J. R., “The mesa problem: Diffusion patterns for $u_t=\nabla\cdot(u^m\nabla u)$ as $m\to+\infty$”, IMA J. Appl. Math., 37 (1986), 147–154 | DOI | MR | Zbl

[22] Hsu S.-Y., “Removable singularities and non-uniqueness of solutions of a singular diffusion equation”, Math. Ann., 325 (2003), 665–693 | DOI | MR | Zbl

[23] Růžička M., Electrorheological Fluids: Modeling and Mathematical Theory, Lect. Notes Math., 1748, Springer, Berlin, 2000 | MR | Zbl

[24] Simon J., “Compact sets in the space $L^p(0,T;B)$”, Ann. Mat. Pura Appl. IV Ser., 146 (1987), 65–96 | DOI | MR | Zbl