Convergence and front position for an FKPP-type free boundary problem
Julien Berestycki, Sarah Penington, Oliver Tough
TL;DR
The paper analyzes a FKPP-type free boundary problem that arises as the hydrodynamic limit of the N-BBM, proving that with suitably fast-decaying initial data the solution converges to the minimal travelling wave. It establishes sharp Bramson-type front-position asymptotics L_t = √2 t − (3/2√2) log t + c + o(1) in the finite initial mass case and extends these sharp results to a β-dependent generalized problem, revealing pulled, pushmi-pullyu, and pushed regimes with precise front-position formulas. A key tool is the Brunet-Derrida relation, linking the initial data's Laplace transform to the front trajectory, which underpins both convergence to Π_min and front-speed characterisation. The work also distinguishes finite vs infinite initial mass cases with rigorous, necessary-and-sufficient conditions for convergence to the minimal wave, and it situates these results within the broader FKPP literature and related particle-system models. Overall, it extends and rigorously confirms non-rigorous physical predictions about front selection, mass effects, and phase transitions in pulled/pushed front dynamics.
Abstract
The free boundary problem\[ \begin{cases} \partial_tu=\frac{1}{2}Δu+u,\quad &t>0, \, x>L_t,\\ u(t,x)=0,\quad &t>0,\, x\le L_t,\\ \int_{L_t}^{\infty}u(t,y)dy=1,\quad &t> 0,\\ u(t,x)dx \to u_0(dx)&\text{weakly as }t\to 0, \end{cases}\] has long been conjectured to be in the universality class of the so-called FKPP reaction-diffusion equation. It appears naturally as the hydrodynamic limit of a branching-selection particle system, the $N$-BBM. In the present work, we show that for any initial condition $u_0(dx)$ that decays fast enough as $x\to\infty$, the solution of the free boundary problem converges to the minimal travelling wave solution. We further show how the decay of the initial condition precisely determines the position of the free boundary $L_t$ at large times $t$, mirroring the celebrated results of Bramson \cite{Bramson1983} in the context of the FKPP equation. Our conditions for convergence to the minimal travelling wave, and for $L_t$ to have the Bramson asymptotics \[ L_t=\sqrt{2}t-\frac{3}{2\sqrt{2}}\log t+c+o(1)\quad\text{as }t\to\infty,\] are necessary and sufficient. We also apply our results to a more general free boundary problem that depends on a parameter $β$, where we see a transition from \emph{pulled} to \emph{pushed} behaviour (with \emph{pushmi-pullyu} behaviour at the critical value of $β$). We obtain analogous sharp conditions for convergence to the minimal travelling wave, along with precise asymptotics for the front position, in each of these regimes. To our knowledge, such necessary and sufficient conditions had not previously been established in the pushmi-pullyu or pushed regimes, even for classical monostable reaction-diffusion equations. Our results prove and extend non-rigorous predictions in the physics literature of the first author, Brunet and Derrida.