De Giorgi type conjecture for reaction-diffusion omega-limit sets

Assume the invasion property, Hypothesis

, holds for some $\rho>0$. Let $u$ solve the homogeneous reaction-diffusion equation with initial datum $u_0=\mathbf{1}_U$, where $U\subset\mathbb{R}^N$ satisfies

d_{\mathcal{H}}(U,U_\rho)<+\infty

$$and the geometric condition$$

. Let $\Omega(u)$ be the omega-limit set of $u$ under time translations. De Giorgi type conjecture. Every function in $\Omega(u)$ is one-dimensional and is either constant or strictly monotone. This extends known asymptotic local planarity results beyond the Fisher–KPP case; the conjecture is open for general nonlinearities satisfying only the invasion property.