Large-population weak-limit conjecture for the finite-population Fisher–KPP equation

Consider the one-dimensional Dirichlet problem

$$\frac{\partial u}{\partial t}=ru(1-u)+\frac{2ak}{ax+b}\frac{\partial u}{\partial x}+k\frac{\partial^2u}{\partial x^2},$$

with the Dirichlet boundary conditions and initial data specified in the paper, and let $u$ denote its solution. Large-population weak-limit conjecture. As $a\to\infty$, the solution $u$ approaches the weak solution

$$u(x)=\begin{cases}1 & \text{if }x=0,\\0 & \text{if }0<x\leq 1.\end{cases}$$

This conjecture proposes the limiting concentration of the solution at the boundary as the population parameter $a$ becomes large; the paper identifies it as a problem for future work.