The logarithmic Bramson correction for Fisher-KPP equations on the lattice ℤ

Abstract

We establish in this paper the logarithmic Bramson correction for Fisher-KPP equations on the lattice Z \mathbb {Z} . The level sets of solutions with step-like initial conditions are located at position c ∗ t − 3 2 λ ∗ ln ⁡ t + O ( 1 ) c_*t-\frac {3}{2\lambda _*}\ln t+O(1) as t → + ∞ t\rightarrow +\infty for some explicit positive constants c ∗ c_* and λ ∗ \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 ∗ c_* uniformly on the positive integers, and that the solutions converge along their level sets to the minimal traveling front for large times.