Past exam of the topics in applied mathematics course of the University of Oxford / 2007 / A / 4 / vi / Solution

Small $L$ makes every nonzero $k_n=n\pi /L$ large, placing them above the finite instability band, where diffusion dominates reaction. The homogeneous zero mode is already stable.

For $0<D<D_c$, the first possible band entry as $L$ increases is $n=1$ crossing $k_1=\sqrt{q_{+}}$. Hence the critical length isAt equality the mode is neutral; it is unstable just above this value, while $\pi /\sqrt{q_{+}}<L<\pi /\sqrt{q_{-}}$. Thus strictly positive growth has infimum $L_c$, with no attained smallest length if “unstable” means $\mathrm{Re}\lambda >0$. The conventional threshold answer is $L_c$. Larger modes cannot enter earlier because their first-entry lengths are $n{L}_c$.