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

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.