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PreprintCC BY 4.0arXiv

The Hamel-Nadirashvili conjecture for the Fisher-KPP equation

  • Hongjun Guo
  • Kelei Wang
  • Maolin Zhou
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Abstract

We resolve the Hamel-Nadirashvili conjecture, which concerns the classification of all [0,1]-valued entire solutions of Fisher-KPP equations with concave C^2 reaction terms in arbitrary space dimension. We prove that every such solution is a C_{loc}^{1,2} limit of solutions generated by finitely many planar traveling fronts, possibly together with a spatially homogeneous solution. Our approach reveals an ordered linear structure underlying the nonlinear solution space, leading to a critical-subcritical spectral decomposition. The proof combines linear reconstruction of subcritical approximants with a measure representation (a kind of nonlinear Martin representation) of critical solutions, obtained through small-amplitude asymptotics and comparison with a radially symmetric entire solution.

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Cite

BibTeXRIS
BibTeX
@misc{guo2026hamelnadirashvili,
  title = {{The Hamel-Nadirashvili conjecture for the Fisher-KPP equation}},
  author = {Hongjun Guo and Kelei Wang and Maolin Zhou},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.27896},
  eprint = {2610.27896},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.27896},
  note = {CC BY 4.0},
}
RIS
TY  - UNPB
TI  - The Hamel-Nadirashvili conjecture for the Fisher-KPP equation
AU  - Hongjun Guo
AU  - Kelei Wang
AU  - Maolin Zhou
PY  - 2026
DA  - 2026/10/20
PB  - arXiv
DO  - 10.48550/arxiv.2610.27896
UR  - https://arxiv.org/abs/2610.27896
AB  - We resolve the Hamel-Nadirashvili conjecture, which concerns the classification of all [0,1]-valued entire solutions of Fisher-KPP equations with concave C^2 reaction terms in arbitrary space dimension. We prove that every such solution is a C_{loc}^{1,2} limit of solutions generated by finitely many planar traveling fronts, possibly together with a spatially homogeneous solution. Our approach reveals an ordered linear structure underlying the nonlinear solution space, leading to a critical-subcritical spectral decomposition. The proof combines linear reconstruction of subcritical approximants with a measure representation (a kind of nonlinear Martin representation) of critical solutions, obtained through small-amplitude asymptotics and comparison with a radially symmetric entire solution.
ER  -