The Hamel-Nadirashvili conjecture for the Fisher-KPP equation

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.