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

Juan Luis Vázquez ()
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Juan Luis Vázquez: Univ. Antónoma de Madrid, Dpto de Mathemáticas

A chapter in Nonlinear Evolution Equations and Related Topics, 2003, pp 67-118 from Springer

Abstract: 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.

Keywords: Porous Medium; Initial Data; Asymptotic Behaviour; Cauchy Problem; Uniform Convergence (search for similar items in EconPapers)
Date: 2003
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Persistent link: https://EconPapers.repec.org/RePEc:spr:sprchp:978-3-0348-7924-8_5

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DOI: 10.1007/978-3-0348-7924-8_5

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