Spatially monotone homoclinic orbits in nonlinear parabolic equations of super-fast diffusion type
type: Item
instance of: scholarly article
title: Spatially monotone homoclinic orbits in nonlinear parabolic equations of super-fast diffusion type
zbMATH Open document ID: 1270.35284
author: Michael Winkler
published in: Mathematische Annalen
publication date: 2013-02-01T00:00:00Z
review text: The subject of this article is the question on the existence of smooth homoclinic orbits in nonlinear parabolic equations without any source. More concretely the authors consider positive classical solutions of the nonlinear diffusion equation \[ u_{t}=u^{p}u_{xx}\eqno (1) \] for \(p>2,\) where \(x\) varies over the real line and the time variable \(t\) is allowed to lie in the whole range \((-\infty, \infty).\) After the substitution \(v=u^{1-p}\) it is transformed into the super-fast diffusion equation \(v_{t}=(v^{m-1}v_{x})_{x}\) with parameter \(m=-\frac{1}{p-1}\) located in the interval \((-1,0)\) according to the requirement \(p>2.\) A positive function \(u\in C^{2,1}(\mathbb{R}\times\mathbb{R})\) is a homoclinic orbit of (1) if \(u(\cdot,t)\rightarrow 0\) locally uniformly on \(\mathbb{R}\) at \(t\rightarrow\pm\infty.\) The main result is the following statement: Theorem. Let \(p>2.\) Then there exists a homoclinic orbit of (1). More precisely, (1) possesses a positive classical solution \(u\in C^{\infty}(\mathbb{R}\times\mathbb{R})\) such that for all \(y\in\mathbb{R}\) one can find \(C>0\) such that \(\|u(\cdot,t)\|_{L^{\infty}((-\infty,y))}\leq Ct^{-\frac{1}{p}}(\ln t)^{\frac{2}{p}}\) for all \(t\geq 2\) and \(\|u(\cdot,t)\|_{L^{\infty}((-\infty,y))}\leq C(-t)^{-\frac{1}{p}}\ln(-t)\) for all \(t\leq -2.\) Moreover, this solution can be constructed in such a way that \(u_{x}>0\) holds in \(\mathbb{R}\times\mathbb{R},\) and that \(u(x,t)\rightarrow 0\) as \(x\rightarrow -\infty,\) even uniformly with respect to \(t\in\mathbb{R},\) that is, \(\|u\|_{L^{\infty}((-\infty,y)\times\mathbb{R})}\rightarrow 0\) as \(y\rightarrow -\infty.\)
reviewed by: Svetlana A. Grishina
Mathematics Subject Classification ID: 35K65
Mathematics Subject Classification ID: 37C29
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35K59
zbMATH DE Number: 6133900
zbMATH Keywords: positive classical solutions
MaRDI profile type: Publication
full work available at URL: s00208-012-0795-z
OpenAlex ID: W2013845882
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DOI: 10.1007/S00208-012-0795-Z
description: scientific article; zbMATH DE number 6133900