Whole-space eventual log-concavity and level-set geometry for the multidimensional Fisher-KPP equation
Abstract
We study eventual strict log-concavity and level-set geometry for the multidimensional Fisher-KPP equation with nontrivial, nonnegative, smooth compactly supported initial data, without assuming radial symmetry or convexity of the initial superlevel sets. Our main result is that there exist T>1 and c>0 such that Dx2log u(t,x)≤ −(c/t)e−β*tId for all t≥ T and x∈ℝN, where β*=2(√2−1). Consequently, after a single waiting time, u(t,·) has exactly one critical point, a nondegenerate global maximizer, and every level set {u(t,·)=λ} with 0<λ<maxx u(t,x) is a smooth closed strictly convex hypersurface with positive principal curvatures. For levels in any fixed closed interval K⊂(0,1), the level sets are radial graphs ρ=Rλ(t,ω), with every principal curvature bounded below by (1−Clog t/t)/(2t) for all sufficiently large t. Unlike log-concavity, ordinary concavity exhibits a threshold determined by the inflection level of the critical travelling wave: for each fixed level above it, u is eventually strictly concave on its superlevel set, whereas below it concavity eventually fails. Our proof proceeds by establishing estimates in four overlapping spatial regions. In the fixed-front region, we combine a one-sided estimate for second angular derivatives with radial and mixed derivative bounds to obtain curvature control without differentiating the Lipschitz directional phase. In the remaining regions, absorptive comparison in the invaded interior, leading-edge barriers and blow-up arguments through the diffusive scale, and Gaussian amplitude estimates combined with parabolic regularity in the exterior yield the required logarithmic-Hessian bounds.
Related
- Large time behaviour in the multidimensional nonlocal Fisher-KPP equation
- Preservation of Log-Concavity and Convergence of Wasserstein-Fisher-Rao Gradient Flows
- Conditional Fisher-Information Central Limit Theorems under Log-Concavity with Information-Theoretic Consequences
- Singularities of Multidimensional Item Response Models and the Rank of the Fisher Information Matrix
- The Hamel-Nadirashvili conjecture for the Fisher-KPP equation
BibTeX
@misc{liang2026wholespace,
title = {{Whole-space eventual log-concavity and level-set geometry for the multidimensional Fisher-KPP equation}},
author = {Xing Liang and Ziqi Zhen},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.30702},
eprint = {2610.30702},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.30702},
}
RIS
TY - UNPB
TI - Whole-space eventual log-concavity and level-set geometry for the multidimensional Fisher-KPP equation
AU - Xing Liang
AU - Ziqi Zhen
PY - 2026
DA - 2026/10/22
PB - arXiv
DO - 10.48550/arxiv.2610.30702
UR - https://arxiv.org/abs/2610.30702
AB - We study eventual strict log-concavity and level-set geometry for the multidimensional Fisher-KPP equation with nontrivial, nonnegative, smooth compactly supported initial data, without assuming radial symmetry or convexity of the initial superlevel sets. Our main result is that there exist T>1 and c>0 such that D_x^2log u(t,x)≤ −(c/t)e^−β_*tId for all t≥ T and x∈ℝ^N, where β_*=2(√2−1). Consequently, after a single waiting time, u(t,·) has exactly one critical point, a nondegenerate global maximizer, and every level set {u(t,·)=λ} with 0<λ<max_x u(t,x) is a smooth closed strictly convex hypersurface with positive principal curvatures. For levels in any fixed closed interval K⊂(0,1), the level sets are radial graphs ρ=R_λ(t,ω), with every principal curvature bounded below by (1−Clog t/t)/(2t) for all sufficiently large t. Unlike log-concavity, ordinary concavity exhibits a threshold determined by the inflection level of the critical travelling wave: for each fixed level above it, u is eventually strictly concave on its superlevel set, whereas below it concavity eventually fails. Our proof proceeds by establishing estimates in four overlapping spatial regions. In the fixed-front region, we combine a one-sided estimate for second angular derivatives with radial and mixed derivative bounds to obtain curvature control without differentiating the Lipschitz directional phase. In the remaining regions, absorptive comparison in the invaded interior, leading-edge barriers and blow-up arguments through the diffusive scale, and Gaussian amplitude estimates combined with parabolic regularity in the exterior yield the required logarithmic-Hessian bounds.
ER -