A note of the convergence of the Fisher-KPP front centred around its $\alpha$-level

Abstract

We consider the solution $u(x,t)$ of the Fisher-KPP equation $\partial_t u=\partial_x^2u+u-u^2$ centred around its $\alpha$-level $\mu_t^{(\alpha)}$ defined as $u(\mu_t^{(\alpha)},t)=\alpha$. It is well known that for an initial datum that decreases fast enough, then $u(\mu_t^{(\alpha)}+x,t)$ converges as $t\to\infty$ to the critical travelling wave. We study in this paper the speed of this convergence and the asymptotic expansion of $\mu_t^{(\alpha)}$ for large~$t$. It is known from Bramson that for initial conditions that decay fast enough, one has $\mu_t^{(\alpha)}=2t-(3/2)\ln t+\text{Cste}+o(1)$. Work is under way \cite{nrr} to show that the $o(1)$ in the expansion is in fact a $k^{(\alpha)}/\sqrt t+\mathcal O(t^{\epsilon-1})$ for any $\epsilon>0$ for some $k^{(\alpha)}$, where it is not clear at this point whether $k^{(\alpha)}$ depends or not on $\alpha$. We show that, unless the time derivative of $\mu_t^{(\alpha)}$ has a very unexpected behaviour at infinity, the coefficient $k^{(\alpha)}$ does not, in fact, depend on $\alpha$. We also conjecture that, for an initial condition that decays fast enough, one has in fact $\mu_t^{(\alpha)}=2t-(3/2)\ln t+\text{Cste}-(3\sqrt\pi)/\sqrt t+g (\ln t)/t +o (1/t)$ for some constant~$g$ which does not depend on $\alpha$.