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

Abstract

Motivated by the study of branching particle systems with selection, we establish global existence for the solution $(u,\mu )$ of the free boundary problem \[ \begin{cases} \partial_t u =\partial^2_{x} u +u & \text{for $t>0$ and $x>{\mu }_t$,}\\ u(x,t)=1 &\text{for $t>0$ and $x\le {\mu }_t$}, \\ \partial_x u(μ_t,t)=0 & \text{for $t>0$}, \\ u(x,0)=v(x) &\text{for $x\in \mathbb{R}$}, \end{cases} \] 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.

References in corpus (4)

• Front propagation into unstable states: Universal algebraic convergence towards uniformly translating pulled fronts
• Exact solution and precise asymptotics of a Fisher-KPP type front
• An exactly solvable travelling wave equation in the Fisher-KPP class
• A free boundary problem in biological selection models

Cited by in corpus (4)

• F-KPP Scaling limit and selection principle for a Brunet-Derrida type particle system
• Branching random walk with non-local competition
• Travelling waves, blow-up and extinction in the Fisher-Stefan model
• Rank Dependent Branching-Selection Particle Systems