= 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 is
$$
\boxed{L_c=\frac\pi{\sqrt{q_+}}
=\pi\sqrt{\frac{2D}{1/2-D+\sqrt{D^2-5D+1/4}}}.}
$$
At 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 $\operatorname{Re}\lambda>0$. The conventional threshold answer is $L_c$. Larger modes cannot enter earlier because their first-entry lengths are $nL_c$.
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