Free boundary problems and biological systems with selection rules

Abstract

We prove the local existence for classical solutions of a free boundary problem which arises in one of the biological selection models proposed by Brunet and Derrida (Phys Rev E 3(56):2597–2604, 1997). The problem we consider describes the limit evolution of branching Brownian particles on the line with death of the leftmost particle at each creation time as studied in De Masi et al. (Hydrodynamics of the N-BBM process, arXiv:1705.01825, 2017). We extensively use results in Cannon (The one-dimensional heat equation, Addison-Wesley Publishing Company, Boston 1984) and Fasano (Mathematical models of some diffusive processes with free boundaries, SIMAI e-Lecture Notes, 2008).