Skip to Main Content

Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

What is MCQ? The Mathematical Citation Quotient (MCQ) measures journal impact by looking at citations over a five-year period. Subscribers to MathSciNet may click through for more detailed information.


The asymptotic behavior of gas in an $n$-dimensional porous medium
HTML articles powered by AMS MathViewer

by Avner Friedman and Shoshana Kamin PDF
Trans. Amer. Math. Soc. 262 (1980), 551-563 Request permission


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 $V(|x|, t)$ which is invariant by similarity transformations and which takes the initial values $\delta (x)I (\delta (x) =$ the Dirac measure) in the distribution sense.
    D. G. Aronson and L. A. Peletier, Large time behavior of solutions of the porous medium equation in bounded domains (to appear).
  • G. I. Barenblatt, Podobie, avtomodel′nost′, promezhutochnaya asimptotika, “Gidrometeoizdat”, Leningrad, 1978 (Russian). Teoriya i prilozheniya k geofizicheskoÄ­ gidrodinamike. [Theory and applications to geophysical hydrodynamics]. MR 556235
  • Ph. Benilan, OpĂ©rateurs accretifs et semigroupes dans les espaces ${L^p}\,(1\, \leqslant \,p\, \leqslant \infty )$ (to appear). H. Brezis and M. G. Crandall, Uniqueness of solutions of the initial-value problem for ${u_t}\, - \,\Delta \varphi (u)$ (to appear).
  • Luis A. Caffarelli and Avner Friedman, Continuity of the density of a gas flow in a porous medium, Trans. Amer. Math. Soc. 252 (1979), 99–113. MR 534112, DOI 10.1090/S0002-9947-1979-0534112-2
  • Luis A. Caffarelli and Avner Friedman, Regularity of the free boundary of a gas flow in an $n$-dimensional porous medium, Indiana Univ. Math. J. 29 (1980), no. 3, 361–391. MR 570687, DOI 10.1512/iumj.1980.29.29027
  • C. J. van Duyn and L. A. Peletier, Asymptotic behaviour of solutions of a nonlinear diffusion equation, Arch. Rational Mech. Anal. 65 (1977), no. 4, 363–377. MR 442479, DOI 10.1007/BF00250433
  • S. Kamenomostkaya, On a problem in the theory of filtration, Dokl. Akad. Nauk SSSR 116 (1957), 18-20.
  • S. Kamenomostskaya, The asymptotic behavior of the solution of the filtration equation, Israel J. Math. 14 (1973), 76–87. MR 315292, DOI 10.1007/BF02761536
  • S. Kamin, Similar solutions and the asymptotics of filtration equations, Arch. Rational Mech. Anal. 60 (1975/76), no. 2, 171–183. MR 397202, DOI 10.1007/BF00250678
  • Olga OleÄ­nik, 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 0192205
  • L. A. Peletier, Asymptotic behavior of solutions of the porous media equation, SIAM J. Appl. Math. 21 (1971), 542–551. MR 304894, DOI 10.1137/0121059
  • R. E. Pattle, Diffusion from an instantaneous point source with a concentration-dependent coefficient, Quart. J. Mech. Appl. Math. 12 (1959), 407–409. MR 114505, DOI 10.1093/qjmam/12.4.407
  • E. S. Sabinina, On the Cauchy problem for the equation of nonstationary gas filtration in several space variables, Soviet Math. Dokl. 2 (1961), 166–169. MR 0158190
  • L. Veron, CoercivitĂ© et propriĂ©tĂ©s regularisantes des semi-groupes non linĂ©aires dans les espaces de Banach (to appear).
Similar Articles
  • Retrieve articles in Transactions of the American Mathematical Society with MSC: 35K05, 76S05
  • Retrieve articles in all journals with MSC: 35K05, 76S05
Additional Information
  • © Copyright 1980 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 262 (1980), 551-563
  • MSC: Primary 35K05; Secondary 76S05
  • DOI:
  • MathSciNet review: 586735