Evolution completeness of separable solutions of non‐linear diffusion equations in bounded domains
Victor A. Galaktionov · Mathematical Methods in the Applied Sciences · 2004
Abstract As a basic example, we consider the porous medium equation (m > 1) equation image where Ω ⊂ ℝN 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 ψk ≠ 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 = {ψk} of such non‐linear eigenfunctions follows from the Lusternik–Schnirel'man variational theory from the 1930s. The first similarity pattern t−1/(m−1)ψ0(x), where ψ0 > 0 in Ω, is known to be asymptotically stable as t → ∞ and attracts all nontrivial solutions with u0 ⩾ 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 = {ψk} of eigenfunctions of the Laplacian Δ in L2. The analysis applies to the perturbed PME and to the p‐Laplacian equations of second and higher order. Copyright © 2004 John Wiley & Sons, Ltd.