Multi-interior-spike solutions for the Cahn-Hilliard equation with arbitrarily many peaks
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)].
scientific article; zbMATH DE number 1455254