Motivated by the study of branching particle systems with selection, 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. Recent results of De Masi \textit{ et al.}~\cite{ DeMasi2017a} show that this global solution can be identified with the hydrodynamic limit of the so-called $N$-BBM, { \it i.e.} a branching Brownian motion in which the population size is kept constant equal to $N$ by killing the leftmost particle at each branching event.
Categories: math.AP math.PR