A construction of patterns with many critical points on topological tori. (English) Zbl 1450.35059
Summary: We consider reaction-diffusion equations on closed surfaces in \(\mathbb{R}^3\) having genus 1. Stable nonconstant stationary solutions are often called patterns. The purpose of this paper is to construct closed surfaces together with patterns having as many critical points as one wants.
MSC:
| 35B36 | Pattern formations in context of PDEs |
| 35B35 | Stability in context of PDEs |
| 35K57 | Reaction-diffusion equations |
| 35K58 | Semilinear parabolic equations |
| 35J61 | Semilinear elliptic equations |
| 35P15 | Estimates of eigenvalues in context of PDEs |
| 35B38 | Critical points of functionals in context of PDEs (e.g., energy functionals) |
| 35B20 | Perturbations in context of PDEs |
| 35R01 | PDEs on manifolds |
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