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On existence of the weak solution for nonlinear diffusion equation. (English) Zbl 0741.35031

A rather general parabolic equation is considered which includes the cases of fast and slow diffusion. The existence of weak solutions is established by means of the line method. It consists of discretizing the problem with respect to the time variable. The problem then reduces to an elliptic system which can be solved by means of a variational method. A solution to the original equation is obtained by letting the time steps tend to zero.
Reviewer: C.Bandle (Basel)

MSC:

35K65 Degenerate parabolic equations
35K57 Reaction-diffusion equations
35A15 Variational methods applied to PDEs
65N40 Method of lines for boundary value problems involving PDEs
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References:

[1] H. W. Alt S. Luckhaus: Quasilinear elliptic-parabolic differential equations. Math. Z. 183, 311-341 (1983). · Zbl 0497.35049
[2] J. Kačur: Method of Rothe in evolution equations. Teubner-Texte zur Mathematik, 80, Leipzig, 1985. · Zbl 0582.65084
[3] S. Fučík A. Kufner: Nonlinear Differential Equations. Amsterdam-Oxford-New Nork, Elsevier 1980.
[4] J. Kačur: On boundedness of the weak solution for some class of quasilinear partial differential equations. Časopis pěst. mat. 98 (1973), 43-55.
[5] J. Nečas: Les méthodes directes en théorie des équations elliptiques. Academia, Prague, 1967. · Zbl 1225.35003
[6] J. Filo: On solutions of a perturbed fast diffusion equation. Aplikace matematiky 32, 1987, 364-380. · Zbl 0652.35064
[7] J. L. Lions: Quelques méthodes de résolution des problémes aux limites non linéaires. Russian translation, Moskva 1972. · Zbl 0248.35001
[8] H. Gajewski K. Gröger K. Zacharias: Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen. Akademie-Verlag, Berlin, 1974. · Zbl 0289.47029
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