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

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 $x^{k+1}{e}^{-\hat{I}{»}_{\ast }x}$ and $x^{\hat{\mathbf{I}}½}{e}^{-\hat{I}»x}$ as $x$ tends to infinity, where $0<\hat{I}»<\hat{I}{»}_{\ast }=\sqrt{f^{\prime }(0)}$ and $k,\hat{\mathbf{I}}½\in \mathbb{R}$. Specifically, our results are the following: For the former type initial data, we prove that the position of the level sets is asymptotically $c_{\ast }t+\frac{k}{2\hat{I}{»}_{\ast }}\ln t+\mathcal{O}(1)$ if $k>-3$, is $c_{\ast }t-\frac{3}{2\hat{I}{»}_{\ast }}\ln t+\frac{1}{\hat{I}{»}_{\ast }}\ln \ln t+\mathcal{O}(1)$ if $k=-3$, where $c_{\ast }=2\hat{I}{»}_{\ast }$. In sharp contrast, if $k<-3$ and if $u_0$ belongs to $\mathcal{O}(x^{k+1}{e}^{-\hat{I}{»}_{\ast }x})$ for $x$ large, then the position of the level sets behaves asymptotically like $c_{\ast }t-\frac{3}{2\hat{I}{»}_{\ast }}\ln t+\ddot{I}{ƒ}_{\infty }+o(1)$, with $\ddot{I}{ƒ}_{\infty }\in \mathbb{R}$ depending on the initial condition $u_0$. Regarding the latter type initial data, we show that the level sets behave asymptotically like $ct+\frac{\hat{\mathbf{I}}½}{\hat{I}}»\ln t$ up to $\mathcal{O}(1)$ error in general setting, with $c=\hat{I}»+f^{\prime }(0)/\hat{I}»$. 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.