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Asymptotic behaviour for the porous medium equation posed in the whole space

In: Nonlinear Evolution Equations and Related Topics

Author

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  • Juan Luis Vázquez

    (Univ. Antónoma de Madrid, Dpto de Mathemáticas)

Abstract

This paper is devoted to present a detailed account of the asymptotic behaviour as t → ∞ of the solutions u(x, t) of the equation (0.1) $$ {u_{{t = }}}\Delta ({u^{m}}) $$ with exponent m > 1, a range in which it is known as the porous medium equation written here PME for short. The study extends the well-known theory of the classical heat equation (HE, the case m = 1) into a nonlinear situation, which needs a whole set of new tools. The space dimension can be any integer n ≥ 1. We will also present the extension of the results to exponents m 0} which lives in L 1 (ℝ n ) ∩ L ∞(ℝ n ) and describes the evolution of the process. The solution is not classical for m > 1, but it is proved that there exists a unique weak solution for all m > 0.

Suggested Citation

  • Juan Luis Vázquez, 2003. "Asymptotic behaviour for the porous medium equation posed in the whole space," Springer Books, in: Wolfgang Arendt & Haïm Brézis & Michel Pierre (ed.), Nonlinear Evolution Equations and Related Topics, pages 67-118, Springer.
    • Handle: RePEc:spr:sprchp:978-3-0348-7924-8_5
    DOI: 10.1007/978-3-0348-7924-8_5
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