A model porous medium equation with variable exponent of nonlinearity: Existence, uniqueness and localization properties of solutions

Nonlinear Analysis, Theory, Methods and Applications

Volumn 60, Issue 3, 2005, Pages 515-545

  Antontsev, S N ^a   Shmarev, S I ^a,b  

^a UNIVERSITY OF BEIRA INTERIOR   (Portugal)

^b UNIVERSITY OF OVIEDO   (Spain)

Author keywords

Nonlinear parabolic equations; Nonstandard growth conditions

Indexed keywords

BIBLIOGRAPHIES; CONTINUUM MECHANICS; DATA REDUCTION; EQUATIONS OF STATE; FUNCTIONS; GRADIENT METHODS; LINEAR EQUATIONS; MATHEMATICAL MODELS; MATRIX ALGEBRA; PERTURBATION TECHNIQUES;

FINITE TIME; LIPSCHITZ-CONTINUOUS BOUNDARY; NONLINEAR PARABOLIC EQUATIONS; NONSTANDARD GROWTH CONDITIONS;

NONLINEAR EQUATIONS;

EID: 9644262686     PISSN: 0362546X     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.na.2004.09.026     Document Type: Article

References (28)

1. A. Acerbi, G. Mignione, G. Seregin, Regularity results for a class of parabolic systems related to a class of non-newtonian fluids, Ann. Inst. H. Poincare 21 (6) (2004) 25-60.
2. S.N. Antontsev, Localization of solutions of degenerate equations of continuum mechanics, Akademiya Nauk SSSR, Sibirsk. Otdel., Institut Gidrodinamiki, Novosibirsk, 1986 (in Russian).
3. S.N. Antontsev, J.I. Díaz, A.V. Domanskiǐ, Stability and stabilization of generalized solutions of degenerate problems of two-phase filtration, Dokl. Akad. Nauk 325 (1992) 1151-1155 (English Trans. in Soviet Phys. Dokl. 37(8) (1992) 411-413).
4. S.N. Antontsev, J.I. Díaz, A.V. Domanskiǐ, Stability and stabilization of generalized solutions of degenerate problems of two-phase filtration, Dokl. Akad. Nauk 325 (1992) 1151-1155 (English Trans. in Soviet Phys. Dokl. 37(8) (1992) 411-413).
5. S.N. Antontsev, J.I. Díaz, A.V. Domanskiǐ, Continuous dependence and stabilization of solutions of the degenerate system in two-phase filtration, Dinamika Sploshn. Sredy, Fluids Dynamics Free Boundary 107 (1993) 11-25.
6. S.N. Antontsev, J.I. Díaz, S. Shmarev, Energy methods for free boundary problems: applications to non-linear PDEs and fluid mechanics, Progress in Nonlinear Differential Equations and their Applications, vol. 48, Bikhäuser, Boston, 2002.
7. S.N. Antontsev, A.V. Domanskiǐ, Uniqueness of generalized solutions of degenerate problems of two-phase filtration, Chisl. Metody Mekh. Sploshn. Sredy 15(6) (1984) 15-27 (in Russian).
8. S.N. Antontsev, A.V. Domanskii, V.I. Penkovskii, Filtration in by-well zone of the formation and the well productivity stimulation problems, Lavrentyev Institute of Hydrodynamics, Siberian Branch of the USSR Academy of Science, Novosibirsk, 1989 (in Russian).
9. S.N. Antontsev, G. Gagneux, Petits paramètres et passages à la limite dans les problèmes de filtration diphasique, Progress Partial Differential Equations (Pont-à- Mousson 1997) 1 (1998).
10. S.N. Antontsev, A.V. Kazhikhov, V.N. Monakhov, Boundary Value Problems in Mechanics of Nonhomogeneous Fluids, North-Holland Publishing Co., Amsterdam, 1990 (Translated from original Russian edition: Nauka, Novosibirsk, 1983).
11. S.N. Antontsev, S.I. Shmarev, A generalized porous medium equation with variable exponents of nonlinearity: existence, uniqueness and localization properties of solutions, The Seminar Dedicated to the Memory of I. Petrovskii, Book of Abstracts, Moskow State University, May 16-22, 2004.
12. D.G. Aronson, The porous medium equation, in: Nonlinear Diffusion Problems (Montecatini Terme, 1985), Lecture Notes in Mathematics, vol. 1224, Springer, Berlin, 1986, pp. 1-46.
13. m as m → ∞, Rend. Sem. Mat. Univ. Politec. Torino (1991) 1-13 (Some topics in nonlinear PDEs, Turin, 1989).
14. E. Chasseigne, J.L. Vazquez, Theory of extended solutions for fast-diffusion equations in optimal classes of data. Radiation from singularities, Arch. Ration. Mech. Anal. 164 (2002) 133-187.
15. m ∇u) as m → +∞, IMA J. Appl. Math. 37 (1986) 147-154.
16. D. Gilbarg, N.S. Trudinger, Elliptic partial differential equations of second order, Grundlehren der Mathematischen Wissenschaften, vol. 224, Springer, Berlin, 1977.
17. S.-Y. Hsu, Removable singularities and non-uniqueness of solutions of a singular diffusion equation, Math. Ann. 325 (2003) 665-693.
18. A.S. Kalashnikov, Some problems of the qualitative theory of second-order nonlinear degenerate parabolic equations, Uspekhi Mat. Nauk 42 (1987) 135-176, 287.
19. A.S. Kalashnikov, Nonlinear effects in heat propagation in media with sources and sinks that are close to linear. Differentsial'nye Uravneniya 31 (2) (1995) 277-288, 366.
20. A.S. Kalashnikov, On some problems in the nonlinear theory of heat conduction with data containing a small parameter in the exponents, Zh. Vychisl. Mat. i Mat. Fiz. 35 (7) (1995) 1077-1094.
21. A.S. Kalashnikov, On some nonlinear problems in mathematical physics with exponents that are close to critical, Trudy Sem imeni I. G. Petrovskogo 19 (1996) 73-98, 346.
22. O.A. Ladyženskaja, V.A. Solonnikov, N.N. Ural'tseva, Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, vol. 23, American Mathematical Society, Providence, RI, 1967 (Translated from the Russian by S. Smith).
23. O.A. Ladyženskaja, N.N. Ural'ceva, Linear and Quasilinear Elliptic Equations, Academic Press, New York, 1968. (Translated from the Russian by Scripta Technica, Inc.).
24. A. Rodriguez, J.L. Vázquez, Non-uniqueness of solutions of nonlinear heat equations of fast diffusion type, Ann. Inst. H. Poincaré Anal. Non Linéaire 12 (1995) 173-200.
25. M. Růžička, Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Mathematics, vol. 1748, Springer, Berlin, 2000.
26. E.S. Sabinina, On a class of non-linear degenerate parabolic equations, Dokl. Akad. Nauk SSSR 143 (1962) 794-797.
27. p(0, T; B), Ann. Mat. Pura Appl. 146 (4) (1987) 65-96.
28. Y.B. Zel'dovič, A.S. Kompaneec, On the theory of propagation of heat with the heat conductivity depending upon the temperature, In: Collection in Honor of the Seventieth Birthday of Academician A. F. loffe, Izdat. Akad. Nauk SSSR, Moscow, 1950, 61-71.