Refined long time asymptotics for Fisher-KPP fronts

Abstract

We study the one-dimensional Fisher-KPP equation, with an initial condition $u_0(x)$ that coincides with the step function except on a compact set. A well-known result of M. Bramson states that, as $t\to+\infty$, the solution converges to a traveling wave located at the position $X(t)=2t-(3/2)\log t+x_0+o(1)$, with the shift $x_0$ that depends on $u_0$. U. Ebert and W. Van Saarloos have formally derived a correction to the Bramson shift, arguing that $X(t)=2t-(3/2)\log t+x_0-3\sqrt{\pi}/\sqrt{t}+O(1/t)$. Here, we prove that this result does hold, with an error term of the size $O(1/t^{1-\gamma})$, for any $\gamma>0$. The interesting aspect of this asymptotics is that the coefficient in front of the $1/\sqrt{t}$-term does not depend on $u_0$.