Free boundary problems and biological systems with selection rules (Q2285066)

The main goal of this paper is to prove local existence for classical solutions of a free boundary problem which arises in one of the biological selection models proposed by \textit{E. Brunet} and \textit{B. Derrida} [``Shift in the velocity of a front due to a cutoff'', Phys. Rev. E 56, No. 3, 2597--2604 (1997; \url{doi:10.1103/PhysRevE.56.2597})]. Uniqueness follows from [\textit{A. De Masi} et al. ``Hydrodynamics of the \(N\)-BBM process'', in: Stochastic dynamics out of equilibrium. Cham: Springer. 523--549 (2017)]. The problem considered describes the limit evolution of branching Brownian particles on the line with death of the utmost particle at each creation time, as studied in [loc. cit.]. The main ingredients in the proof are results in [\textit{J. R. Cannon}, The one-dimensional heat equation. Menlo Park, California etc.: Addison-Wesley Publishing Company; Cambridge etc.: Cambridge University Press (1984; Zbl 0567.35001)] and [\textit{A. Fasano}, Mathematical models of some diffusive processes with free boundaries. Rosario: Universidad Austral, Departamento de Matemática (2005; Zbl 1122.35001)]. More especifically, fix \(b\in\mathbb{R}\), \(b>0\), \(\rho_0\in C^2_c([b,\infty))\) with \(\rho_0(b)=0\), \(\frac{d}{dx}\rho_0(b)=2\) and \(\int_b^{\infty}\rho_0(x)dx=1\). The problem at hand consists of finding a continuous curve \(X_t, \ t\ge 0\), with \(X_0=b\) and \(\rho(x,t),x\ge X_t\), \(t\ge 0\), such that \[ \rho_t(x,t)=\frac{1}{2}\rho_{xx}(x,t)+\rho(x,t), \ \ \text{ if } X_t < x, \ t>0 \] \[ \rho(X_t,t)=0, \ \ \text{ if } t \ge 0, \] \[ \rho(x,0)=\rho_0(x), \ \ \text{ if } b\le x, \] \[ \int_{X_t}^{\infty}\rho(x,t)dx=\int_b^{\infty}\rho_0(x)dx=1, \ \ \text{ if } t>0. \] A pair \((X,\rho)\) is a classical solution of the above problem in the time interval \([0,T]\), \(T>0\), if the following hold: \(X\in C^1([0,T])\), \(X_0=b\), \(\rho\in C(\overline{D_{X,T}})\cap C^{2,1}(D_{X,T})\), where \(D_{X,T}=\{(x,t) \ | \ X_t < x, \ 0 < t < T\}\), and \((X,\rho)\) satisfies the above equations. The author proves existence of \(T>0\) so that a classical solution in the time \([0,T]\) exists. The key idea is to define \(v(x,t)=e^{-t}\rho_x(x,t)\) and to reduce the existence of a solution to the problem satisfied by \((X,v)\) to a fixed point problem.