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Transactions of the American Mathematical Society

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The asymptotic behavior of gas in an $ n$-dimensional porous medium


Authors: Avner Friedman and Shoshana Kamin
Journal: Trans. Amer. Math. Soc. 262 (1980), 551-563
MSC: Primary 35K05; Secondary 76S05
DOI: https://doi.org/10.1090/S0002-9947-1980-0586735-0
MathSciNet review: 586735
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Abstract: Consider the flow of gas in an n-dimensional porous medium with initial density $ {u_0}(x)\, \geqslant \,0$. The density $ u(x,\,t)$ then satisfies the nonlinear degenerate parabolic equation $ {u_t}\, = \,\Delta {u^m}$ where $ m\, > \,1$ is a physical constant. Assuming that $ I\, \equiv \,\int {\,{u_0}(x)} dx\, < \,\infty $ it is proved that $ u(x,\,t)$ behaves asymptotically, as $ t\, \to \,\infty $, like the special (explicitly given) solution $ V(\vert x\vert,\,t)$ which is invariant by similarity transformations and which takes the initial values $ \delta (x)I\,(\delta (x)\, = \,$ the Dirac measure) in the distribution sense.


References [Enhancements On Off] (What's this?)

  • [1] D. G. Aronson and L. A. Peletier, Large time behavior of solutions of the porous medium equation in bounded domains (to appear).
  • [2] G. I. Barenblatt, Similarity, automodelling, intermediate asymptotics; theory and applications to geophysical hydro-dynamics, Gidrometeoizdat, Leningrad, 1978; English transl., Similarity, self-similarity and intermediate asymptotics, Consultants Bureau, New York and London, 1979. MR 556235 (82c:76001a)
  • [3] Ph. Benilan, Opérateurs accretifs et semigroupes dans les espaces $ {L^p}\,(1\, \leqslant \,p\, \leqslant \infty )$ (to appear).
  • [4] H. Brezis and M. G. Crandall, Uniqueness of solutions of the initial-value problem for $ {u_t}\, - \,\Delta \varphi (u)$ (to appear).
  • [5] L. A. Caffarelli and A. Friedman, Continuity of the density of a gas flow in a porous medium, Trans. Amer. Math. Soc. 252 (1979), 99-113. MR 534112 (80i:35090)
  • [6] -, Regularity of the free boundary of a gas flow in an n-dimensional porous medium, Indiana Univ. Math. J. 29 (1980), 361-391. MR 570687 (82a:35096)
  • [7] C. J. Van Duyan and L. A. Peletier, Asymptotic behavior of solution of nonlinear diffusion equation, Arch. Rational Mech. Anal. 65 (1977), 363-377. MR 0442479 (56:861)
  • [8] S. Kamenomostkaya, On a problem in the theory of filtration, Dokl. Akad. Nauk SSSR 116 (1957), 18-20.
  • [9] -, The asymptotic behavior of the solution of the filtration equation, Israel J. Math. 14 (1973), 76-87. MR 0315292 (47:3841)
  • [10] S. Kamin, Similar solutions and the asymptotics of filtration equations, Arch. Rational Mech. Anal. 60 (1976), 171-183. MR 0397202 (53:1061)
  • [11] O. A. Oleinik, On some degenerate quasilinear parabolic equations, Seminari dell' Istituto Nazionale di Alta Matematica 1962-1963, Oderisi, Gubbio, 1964, pp. 355-371. MR 0192205 (33:432)
  • [12] L. A. Peletier, Asymptotic behavior of the solutions of the porous media equation, SIAM J. Appl. Math. 21 (1971), 542-551. MR 0304894 (46:4026)
  • [13] R. E. Prattle, Diffusion from an instantaneous point source with concentration-dependent coefficient, Quart. J. Mech. Appl. Math. 12 (1959), 407-409. MR 0114505 (22:5326)
  • [14] E. S. Sabinina, On the Cauchy problem for the equation of non-stationary gas filtration in several space variables, Dokl. Akad. Nauk SSSR 136 (1961), 1034-1037. MR 0158190 (28:1416)
  • [15] L. Veron, Coercivité et propriétés regularisantes des semi-groupes non linéaires dans les espaces de Banach (to appear).

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DOI: https://doi.org/10.1090/S0002-9947-1980-0586735-0
Article copyright: © Copyright 1980 American Mathematical Society

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