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

DOI10.1088/1361-6544/AB25AFzbMATH Open1420.35474arXiv1805.03702OpenAlexW2802138146MaRDI QIDQ5195024

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Publication date: 17 September 2019

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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 [ �egin{cases} partial_t u =partial^2_{x} u +u & ext{for $t>0$ and $x>m{u}_t$,}\ u(x,t)=1 & ext{for $t>0$ and $xleqm{u}_t$}, \ partial_x u(mu_t,t)=0 & ext{for $t>0$}, \ u(x,0)=v(x) & ext{for $xinmathbbR$}, end{cases} ] when the initial condition $v:mathbbRo[0,1]$ is non-increasing with $v(x)o0$ as $xoinfty$ and $v(x)o1$ as $xo-infty$. We construct the solution as the limit of a sequence $(u_n{)}_{nge1}$, 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 extit{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.

Full work available at URL: https://arxiv.org/abs/1805.03702

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