Archive for Rational Mechanics and Analysis

, Volume 60, Issue 2, pp 171–183 | Cite as

Similar solutions and the asymptotics of filtration equations

  • S. Kamin Kamenomostskaya
Article
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Abstract

In this paper we consider the asymptotic behavior of solutions of the quasilinear equation of filtration as t→∞. We prove that similar solutions of the equation ut = (uλ)xx asymptotically represent solutions of the Cauchy problem for the full equation ut = [φ(u)]xx if ⩼φ(u) is “close” to uλ for small u.

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References

  1. 1.
    Barenblatt, G. I., & Ya. B. Zel'dovich, Self-similar solutions as intermediate asymptotics. Annual Review of Fluid Mech. V. 4, 285–312 (1972).Google Scholar
  2. 2.
    Il'in, A. M., A. S. Kalashnikov, & O. A. Oleinik, Linear second order parabolic equations. Usp. Math. Nauk SSSR, 17, no. 3, 3–146 (1962) (Russian).Google Scholar
  3. 3.
    Kamenomostskaya, S., The asymptotic behaviour of the solution of the filtration equation. Israel J. of Math. 14, 1, 76–87 (1973).Google Scholar
  4. 4.
    Kamin (Kamenomostskaya), S., Some estimates for solutions of the Cauchy problem for equations of a nonstationary equations. J. Diff. Eq. (to appear).Google Scholar
  5. 5.
    Oleinik, O. A., A. S. Kalashnikov, & Chzhou, Yui-Lin, The Cauchy problem and boundary problems for equations of the type of non-stationary filtration. Izv. Akad. Nauk SSSR, Ser. Mat. 22, 667–704 (1958) (Rus.)Google Scholar
  6. 6.
    Peletier, L. A., On the asymptotic behaviour of velocity profiles in laminar boundary layers. Arch. Rational Mech. Anal. 45, 110–119 (1972).Google Scholar
  7. 7.
    Peletier, L. A., Asymptotic behaviour of temperature profiles of a class of non-linear heat conduction problems. Quart. Journ. Mech. Appl. Math., 23, 441–447 (1970).Google Scholar
  8. 8.
    Peletier, L. A., Asymptotic behaviour of solutions of the porous media equation. SIAM J. Appl. Math., V. 21, No. 4 (1971).Google Scholar
  9. 9.
    Serrin, J., Asymptotic behaviour of velocity profiles in the Prandtl boundary layer theory. Proc. R. Soc. A299, 491–507 (1967).Google Scholar
  10. 10.
    Sobolev, S. L., Applications of Functional Analysis in Math. Physics. Amer. Math. Soc. Translations 7. Providence, R.I. (1963).Google Scholar
  11. 11.
    Zel'dovich, I. B., & A. S. Kompaneez, On the theory of heat propagation with heat conduction depending on temperature: Lectures dedicated on the 70th Anniversary of A. F. Joffe. Akad. Nauk SSSR, 1950 (Russian).Google Scholar

Copyright information

© Springer-Verlag 1976

Authors and Affiliations

  • S. Kamin Kamenomostskaya
    • 1
  1. 1.Department of Mathematical SciencesTel-Aviv UniversityIsrael

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