Multi-interior-spike solutions for the Cahn-Hilliard equation with arbitrarily many peaks
type: Item
instance of: scholarly article
title: Multi-interior-spike solutions for the Cahn-Hilliard equation with arbitrarily many peaks
zbMATH Open document ID: 0952.35029
author: Matthias Winter
author: Juncheng Wei
published in: Calculus of Variations and Partial Differential Equations
publication date: 2001-01-04T00:00:00Z
full work available at URL: http://bura.brunel.ac.uk/handle/2438/518
review text: The existence of spike-layer solutions and the location and profile of the peaks are considered for the stationary solutions of the Cahn-Hillard equation. Such solutions satisfy the nonlinear elliptic equation \[ \varepsilon ^{2}\Delta u-f(u)=\lambda _{\varepsilon }\quad \text{in } \Omega ,\quad \frac{\partial }{\partial v}u=0\quad \text{on }\partial \Omega , \qquad \int_{\Omega }udx=m|\Omega |, \] where \(\Omega \) is a bounded smooth domain in \(\mathbb{R}^{n}\), \(\varepsilon \) is sufficiently small, \(\lambda _{\varepsilon }\) is a constant and the nonlinear function \(f\) is the first derivative of a free energy density. For any positive integer \(K\), an interior \(K-\)spike solution is constructed, which possesses peaks with maximal possible distance from the boundary and from one another. It implies the existence of an interior \(K-\)peaked solution. The paper is a continuation of some recent authors on the Cahn-Hillard equation and spike-layer solutions. See for instance \textit{J. Wei} and \textit{M. Winter} [Ann. Inst. Henri Poincaré, Anal. Non Lineaire 15, No. 4, 459-492 (1998; Zbl 0910.35049)].
reviewed by: Mervan Pašić
Mathematics Subject Classification ID: 35J40
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35B45
zbMATH DE Number: 1455254
zbMATH Keywords: spike-layer solutions
zbMATH Keywords: Cahn-Hillard equation
zbMATH Keywords: nonlinear elliptic equations
zbMATH Keywords: Lyapunov-Schmidt method
MaRDI profile type: Publication
OpenAlex ID: W1979322551
Wikidata QID: Q115387752
DOI: 10.1007/S005260050153
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description: scientific article; zbMATH DE number 1455254