Abstract
As a basic example, we consider the porous medium equation (m > 1)
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where Ω ⊂ ℝ^N^ is a bounded domain with the smooth boundary ∂Ω, and initial data $u_0 \thinspace \varepsilon L^{\infty} \cap L^{1}$. It is well‐known from the 1970s that the PME admits separable solutions $u_{k}(x,t) = t^{-1/(m-1)} \psi_{ k}(x), , k = 0,1,2,\ldots,$, where each ψ~k~ ≠ 0 satisfies a non‐linear elliptic equation $\Delta (|\psi_{ k}|^{m-1} \psi_k)+ 1/(m-1) \psi_k = 0, in ,\Omega, \psi_{ k} = 0, on , \partial \Omega$. Existence of at least a countable subset Φ = {ψ~k~} of such non‐linear eigenfunctions follows from the Lusternik–Schnirel'man variational theory from the 1930s. The first similarity pattern t^−1/(m−1)^ψ~0~(x), where ψ~0~ > 0 in Ω, is known to be asymptotically stable as t → ∞ and attracts all nontrivial solutions with u~0~ ⩾ 0 (Aronson and Peletier, 1981).
We show that if Φ is discrete, then it is evolutionary complete, i.e. describes the asymptotics of arbitrary global solutions of the PME. For m = 1 (the heat equation), the evolution completeness follows from the completeness‐closure of the orthonormal subset Φ = {ψ~k~} of eigenfunctions of the Laplacian Δ in L^2^. The analysis applies to the perturbed PME and to the p‐Laplacian equations of second and higher order. Copyright © 2004 John Wiley & Sons, Ltd.