2022/06/09 by Besse, Christophe, Faye, Grégory, Roquejoffre, Jean-Michel +1
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2206.04358
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-(3)/(2λ_*)ln t+O(1) as t→+∞ for some explicit positive constants c_* and λ_*. 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.