 Solution

Put $a=f^{\prime }(0)>0$. Linearizing the morphogen reaction-diffusion equation at $C=0$ givesSeek separated modes $C(x,t)=X(x)e^{\lambda t}$. ThenThe mixed Dirichlet-Neumann modes on an interval areIndeed, the Dirichlet condition selects sine functions, while the Neumann condition requires $\cos (k_{n}L)=0$. Their growth rates areThe largest growth rate is the lowest mode,Linear stability requires ${\lambda }_0<0$, after which all higher modes also decay. Therefore the critical length for a linearly growing morphogen isEquality gives a neutral lowest mode, while a larger domain is linearly unstable.