Multiple spike layers in the shadow Gierer-Meinhardt system: Existence of equilibria and the quasi-invariant manifold
The author considers the problem: \(-\Delta u+ \varepsilon^{-2}(u- f(u))= 0\), in \(\Omega\), \(\partial u/\partial n= 0\) on \(\partial\Omega\), \(u>0\) in \(\Omega\), where \(\varepsilon\) is a small parameter. \(\Omega\subset \mathbb{R}^d\) is a bounded domain with \(C^3\) boundary and \(f\in C^{1,1}(\mathbb{R})\) satisfies \(f\equiv 0\) in \((-\infty, 0)\), \(f(u)= O(u^p)\) as \(u\to\infty\), and \(1< p<(d+ 2)/(d- 2)\), (if \(d=2\), then \(1< p<\infty\)). Additionally, it assumes that there exists \(q>1\) such that for \(u>0\) the function \(f(u)/u^q\) is nondecreasing.
scientific article; zbMATH DE number 1425216
scientific article