General periodic logarithmic front-position conjecture

Establish that, for the spatially periodic lattice Fisher–KPP equation with finitely supported nonzero initial data, the level-set position in any observation direction e and within any fixed-width lattice tube satisfies R_{\theta,e,\rho}(t)=w(e)t-\frac{d+2}{2p_e\cdot e}\log t+O(1) as t\to\infty, for every \theta\in(0,1), e\in\mathbb S^{d-1}, and \rho\geq\sqrt d.

Background

The paper studies the lattice Fisher–KPP equation u_t=\Delta_{\mathbb Zd}u+F(x,u) with spatially periodic KPP reaction, localized initial data, and a directional spreading geometry determined by the Perron eigenvalue H(p) of the tilted finite-cell operator. For an observation direction e, w(e) is the leading radial spreading speed, p_e is the associated critical spectral vector, and p_e\cdot e determines the direction-dependent logarithmic coefficient.

The conjecture concerns the full spatially periodic case, in which the linear growth rate may vary in every coordinate and the propagation direction need not be a coordinate axis. Theorem 2.1 establishes the corresponding logarithmic upper bound. Theorem 2.2 proves a matching lower bound only in the axial setting when the linear growth rate is independent of the propagation coordinate. The paper reduces the unresolved general lower bound to two estimates: a positive near-boundary lower bound for a time-changed killed kernel and a uniform bound on the accumulated nonlinear loss represented by the Dirichlet potential J_v.

References

Under eq:periodic-kpp--eq:u0, for every fixed $\theta\in(0,1)$, $e\in\mathbb S{d-1}$ and $\rho\geq\sqrt d$, \begin{equation}\label{eq:general-target} R_{\theta,e,\rho}(t)=w(e)t- \frac{d+2}{2p_e\cdot e}\log t+O(1). \end{equation}

eq:general-target:

Rθ,e,ρ(t)=w(e)t−d+22pe⋅elog⁡t+O(1).R_{\theta,e,\rho}(t)=w(e)t- \frac{d+2}{2p_e\cdot e}\log t+O(1).

eq:periodic-kpp:

F(x,⋅)∈C2([0,1]),F(x,0)=F(x,1)=0,a(x):=Fs(x,0)>0,Fs(x,1)<0,0<F(x,s)≤a(x)s(0<s<1).\begin{gathered} F(x,\cdot)\in C^2([0,1]),\quad F(x,0)=F(x,1)=0,\\ a(x):=F_s(x,0)>0,\quad F_s(x,1)<0,\\ 0<F(x,s)\leq a(x)s\quad(0<s<1). \end{gathered}

eq:u0:

$0\leq u_0\leq1,\qquad u_0\not\equiv0,\qquad \supp(u_0)\text{ is finite}. $

— Logarithmic upper bounds and axial front selection for spatially periodic lattice Fisher--KPP equations  (2610.06381 - Hu et al., 5 Oct 2026) in Conjecture (Section 8, “A criterion for the general periodic lower bound”)