Global existence for a free boundary problem of Fisher-KPP type

Julien Berestycki, Eric Brunet, Sarah Penington

Published 2018-05-09Version 1

We establish global existence for the solution $(u,\mu)$ of the free boundary problem \[ \begin{cases} \partial_t u =\partial^2_{x} u +u & \text{for $t>0$ and $x>\mu_t$,} u(x,t)=1 &\text{for $t>0$ and $x \leq \mu_t$}, \partial_x u(\mu_t,t)=0 & \text{for $t>0$}, u(x,0)=v(x) &\text{for $x\in \mathbb{R}$}, \end{cases} \] when the initial condition $v:\mathbb{R}\to[0,1]$ is non-increasing with $v(x) \to 0$ as $x\to \infty$ and $v(x)\to 1$ as $x\to -\infty$. We construct the solution as the limit of a sequence $(u_n)_{n\ge 1}$, where each $ u_n$ is the solution of a Fisher-KPP equation with same initial condition, but with a different non-linear term. The existence of a solution of the free boundary problem allows us to strengthen a recent result by De Masi \textit{et al.}~on the hydrodynamic limit of the $N$-BBM.