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 ${\partial }_{t}u={\partial }_x^{2}u+u-u^2$ 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 $\alpha$.

12 pages; 2 figures