Abstract
We formulate the filtration problem with free boundary as a problem with discontinuous nonlinearity for a degenerate elliptic or parabolic system. We prove that a solution of the Dirichlet problem exists in both cases. We study some qualitative properties of these solutions, e.g., the existence of “dead cores”.
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References
K. C. Chang, “The obstacle problem and partial differential equations with discontinuous nonlinearities,”Comm. Pure Appl. Math.,33, No. 2, 117–147 (1980).
C. Bandle, R. P. Sperb, and I. Stakgold, “Diffusion and reaction with monotone kinetics,”Nonlin. Analysis,8, No. 4, 321–334 (1984).
R. H. Martin and M. E. Oxley, “Moving boundaries in reaction-diffusion systems with absorption,”Nonlin. Analysis,14, No. 2, 167–192 (1990).
V. S. Nustrov and A. V. Plastinin, “On a Stephan-type problem of filtration of liquid in a cracked porous seam,”Inzh.-Fiz. Zh.,65, No. 2, 26–28 (1993).
D. Kinderlehrer and G. Stampacchia,An Introduction to Variational Inequalities and Their Applications, Academic Press, New York (1980).
J. L. Lions,Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod Gauthier-Villars, Paris (1969).
J. Hernandez, “Some free boundary problems for predator-prey systems with nonlinear diffusion,”Proc. Symp. Pure Math.,45, 481–488 (1984).
L. Maddalena, “Existence of global solution for reaction-diffusion systems with density dependent diffusion,”Nonlin. Analysis,8, No. 11, 1383–1394 (1984).
O. A. Ladyzhenskaya, V. A. Solonnikov, and N. N. Ural'tseva,Linear and Quasilinear Equations of Parabolic Type [in Russian], Nauka, Moscow (1967).
D. S. Aronson, “The porous medium equation,”Lect. Notes Math.,1224, 1–46 (1986).
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Institute of Applied Mathematics and Mechanics, Ukrainian Academy of Sciences, Donetsk. Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 48, No. 9, pp. 1155–1165, September, 1996.
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Bazalii, B.V., Krasnoshchek, N.V. On one problem with free boundary for a nonlinear system. Ukr Math J 48, 1309–1321 (1996). https://doi.org/10.1007/BF02595354
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DOI: https://doi.org/10.1007/BF02595354