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Transactions of the American Mathematical Society

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The porous medium equation in one dimension


Author: Barry F. Knerr
Journal: Trans. Amer. Math. Soc. 234 (1977), 381-415
MSC: Primary 35K15
DOI: https://doi.org/10.1090/S0002-9947-1977-0492856-3
MathSciNet review: 0492856
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Abstract: We consider a second order nonlinear degenerate parabolic partial differential equation known as the porous medium equation, restricting our attention to the case of one space variable and to the Cauchy problem where the initial data are nonnegative and have compact support consisting of a bounded interval. Solutions are known to have compact support for each fixed time.

In this paper we study the lateral boundary, called the interface, of the support $ P[u]$ of the solution in $ {R^1} \times (0,T)$. It is shown that the interface consists of two monotone Lipschitz curves which satisfy a specified differential equation. We then prove results concerning the behavior of the interface curves as t approaches zero and as t approaches infinity, and prove that the interface curves are strictly monotone except possibly near $ t = 0$. We conclude by proving some facts about the behavior of the solution in $ P[u]$.


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  • [1] D. G. Aronson, Regularity properties of flows through porous media: The interface, Arch. Rational Mech. Anal. 37 (1970), 1-10. MR 41 #656. MR 0255996 (41:656)
  • [2] -, Regularity properties of flows through porous media: A counterexample, SIAM J. Appl. Math. 19 (1970), 299-307. MR 42 #683. MR 0265774 (42:683)
  • [3] -, Regularity properties of flows through porous media, SIAM J. Appl Math. 17 (1969), 461-467. MR 40 #571. MR 0247303 (40:571)
  • [4] G. I. Barenblatt, On one class of exact solutions of the plane one-dimensional problem of unsteady filtration of a gas in a porous medium, Akad. Nauk SSSR Prikl. Mat. Meh. 17 (1953), 739-742. MR 16, 298. MR 0064556 (16:298h)
  • [5] -, On some unsteady motions of a liquid and a gas in a porous medium, Akad. Nauk SSSR Prikl. Mat. Meh. 16 (1952), 67-78. MR 13, 700. MR 0046217 (13:700a)
  • [6] A. Friedman, Partial differential equations of parabolic type, Prentice-Hall, Englewood Cliffs, N.J., 1964. MR 31 #6062. MR 0181836 (31:6062)
  • [7] A. S. Kalašnikov, On the differential properties of generalized solutions of equations of the nonsteady filtration type, Vestnik Moskov. Univ. Ser. I Mat. Meh. 29 (1974), 62-68 = Moscow Univ. Math. Bull. 29 (1974), 48-53. MR 49 #7578. MR 0342834 (49:7578)
  • [8] -, On equations of the nonstationary filtration type in which the perturbation is propagated at infinite velocity, Vestnik. Moscov. Univ. Ser. I Mat. Meh. 27 (1972), 45-49 = Moscow Univ. Math. Bull. 27 (1972), 104-108 (1973). MR 49 #849. MR 0336073 (49:849)
  • [9] -, Formation of singularities in solutions of the equation of nonstationary filtration, Ž. Vyčisl. Mat. i Mat. Fiz. 7 (1967), 440-444. MR 35 #1940. MR 0211058 (35:1940)
  • [10] S. Kamenomostskaya, The asymptotic behavior of the solution of the filtration equation, Israel J. Math. 14 (1973), 76-87. MR 47 #3841. MR 0315292 (47:3841)
  • [11] O. A. Ladyženskaja, V. A. Solonnikov and N. N. Ural'ceva, Linear and quasilinear equations of parabolic type, ``Nauka", Moscow, 1967; English transl., Transl. Math. Monographs, vol. 23, Amer. Math. Soc., Providence, R. I., 1968. MR 39 #3159 a,b. MR 0241822 (39:3159b)
  • [12] M. Muskat, The flow of homogeneous fluids through porous media, McGraw-Hill, New York, 1937.
  • [13] O. A. Oleĭnik, On some degenerate quasilinear parabolic equations, Seminari 1962/63 Anal. Alg. Geom. e Topol., Vol. 1, Ist.Naz. Alta Mat. Ediz. Cremonese, Rome, 1965, pp. 355-371. MR 33 #432. MR 0192205 (33:432)
  • [14] O. A. Oleĭnik, A. S. Kalašnikov and Yui-Lin' Čžou, The Cauchy problem and boundary problems for equations of the type of non-stationary filtration, Izv. Akad. Nauk SSSR Ser. Mat. 22 (1958), 667-704. (Russian) MR 20 #6271. MR 0099834 (20:6271)
  • [15] R. E. Prattle, Diffusion from an instantaneous point source with a concentration-dependent coefficient, Quart. J. Mech. Appl. Math. 12 (1959), 407-409. MR 0114505 (22:5326)
  • [16] M. H. Protter, Properties of solutions of parabolic equations and inequalities, Canad. J. Math. 13 (1961), 331-345. MR 27 #3943. MR 0153982 (27:3943)
  • [17] E. S. Sabinina, On the Cauchy problem for the equation of nonstationary gas filtration in several space variables, Dokl. Akad. Nauk SSSR 136 (1961), 1034-1037 = Soviet Math. Dokl. 2 (1961), 166-169. MR 28 #1416. MR 0158190 (28:1416)

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DOI: https://doi.org/10.1090/S0002-9947-1977-0492856-3
Keywords: Nonlinear degenerate parabolic second order equation, porous medium
Article copyright: © Copyright 1977 American Mathematical Society

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