The logarithmic Bramson correction for Fisher-KPP equations on the lattice $\mathbb{Z}$

Abstract

We establish in this paper the logarithmic Bramson correction for Fisher-KPP equations on the lattice $\Z$. The level sets of solutions with step-like initial conditions are located at position $c_*t-\frac{3}{2\lambda_*}\ln t+O(1)$ as $t\rightarrow+\infty$ for some explicit positive constants $c_*$ and $\lambda_*$. This extends a well-known result of Bramson in the continuous setting to the discrete case using only PDE arguments. A by-product of our analysis also gives that the solutions approach the family of logarithmically shifted traveling front solutions with minimal wave speed $c_*$ uniformly on the positive integers, and that the solutions converge along their level sets to the minimal traveling front for large times.