Sharp asymptotics for the KPP equation with some front-like initial data

TL;DR

This paper rigorously derives the large-time asymptotics of the KPP equation's solutions with specific front-like initial data, extending Bramson's classical results with precise logarithmic corrections and convergence properties.

Contribution

First PDE proof of Bramson's 1983 asymptotics for the KPP equation with detailed asymptotic formulas for various initial data types.

Findings

Asymptotic position of level sets for $x^{k+1}e^{- heta x}$ initial data.

Logarithmic correction terms in level set positions.

Convergence to traveling waves under precise decay conditions.

Abstract

We provide the first PDE proof of the celebrated Bramson's $o(1)$ results in 1983 concerning the large time asymptotics for the KPP equation under front-like initial data of types $x^{k+1}e^{-\lambda_*x}$ and $x^{\boldsymbol{\nu}} e^{-\lambda x}$ as $x$ tends to infinity, where $0<\lambda<\lambda_*=\sqrt{f'(0)}$ and $k, \boldsymbol{\nu}\in\mathbb{R}$. Specifically, our results are the following: For the former type initial data, we prove that the position of the level sets is asymptotically $c_*t+\frac{k}{2\lambda_*}\ln t+\mathcal{O}(1)$ if $k>-3$, is $c_*t-\frac{3}{2\lambda_*}\ln t+\frac{1}{\lambda_*}\ln\ln t+\mathcal{O}(1)$ if $k=-3$, where $c_*=2\lambda_*$. In sharp contrast, if $k<-3$ and if $u_0$ belongs to $\mathcal{O}(x^{k+1}e^{-\lambda_* x})$ for $x$ large, then the position of the level sets behaves asymptotically like $c_*t-\frac{3}{2\lambda_*}\ln t+\sigma_\infty+o(1)$, with $\sigma_\infty\in\mathbb{R}$ depending on the initial condition $u_0$. Regarding the latter type initial data, we show that the level sets behave asymptotically like $ct+\frac{\boldsymbol{\nu}}{\lambda}\ln t$ up to $\mathcal{O}(1)$ error in general setting, with $c=\lambda+f'(0)/\lambda$. Under the $\mathcal{O}(1)$ results, the ``convergence along level sets'' results are also demonstrated. Moreover, we further refine the above $\mathcal{O}(1)$ results to the ``convergence to a traveling wave'' results provided that initial data decay precisely as a multiple of the above decaying rates.