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

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Nonlinear Evolution Equations and Related Topics

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

$$ {u_{{t = }}}\Delta ({u^{m}}) $$
((0.1))

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 < 1 (fast-diffusion equation, Fde). For definiteness we consider the Cauchy Problem posed in Q = ℝn x ℝ+ with initial data

$$ u(x,0) = {u_{0}}(x), x \in {\mathbb{R}^{n}} $$
((0.2))

chosen in a suitable class of functions. In most of the paper we concentrate on the class X 0 of integrable and nonnegative data,

$$ {u_{0}} \in {L^{1}}({\mathbb{R}^{n}}), {u_{0}} \geqslant 0, $$
((0.3))

which is natural on physical grounds as the density or concentration of a diffusion process, the height of a ground-water mound, or the temperature of a hot medium (see a comment on the applications at the end). Consequently, we will deal mostly with nonnegative solutions u(x, t) ≥ 0 defined in Q. An existence and uniqueness theory exists for this problem so that for every data uo we can produce an orbit {u(·,t):t > 0} which lives in L 1 (ℝn) ∩ L ∞(ℝn) 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.

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Authors and Affiliations

  1. Dpto de Mathemáticas, Univ. Antónoma de Madrid, E-29049, Madrid, Spain

    Juan Luis Vázquez

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  1. Juan Luis Vázquez
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Editors and Affiliations

  1. Department of Applied Analysis, University of Ulm, 89069, Ulm, Germany

    Wolfgang Arendt

  2. Analyse Numérique, Université Pierre et Marie Curie, B.C. 187, 4, place Jussieu, 75252, Paris Cedex 05, France

    Haïm Brézis

  3. Department of Mathematics Hill Center, Busch Campus, Rutgers University, 110 Frelinghuysen RD, Piscataway, NJ, 08854, USA

    Haïm Brézis

  4. Campus de Ker Lann, Antenne de Bretagne de l’ENS Cachan, 35170, Bruz, France

    Michel Pierre

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Dédié à la mémoire de Philippe Bénilanami et maître

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Vázquez, J.L. (2003). Asymptotic behaviour for the porous medium equation posed in the whole space. In: Arendt, W., Brézis, H., Pierre, M. (eds) Nonlinear Evolution Equations and Related Topics. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-7924-8_5

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  • DOI: https://doi.org/10.1007/978-3-0348-7924-8_5

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