 Solution

Only the Laplacian eigenfunctions whose eigenvalues lie in the intersection of the unstable band $(q_{-},q_{+})$ with the allowed spectrum can grow. A lower bound $k^2>k_0^2$ removes all unstable modes if $k_0^2\ge q_{+}$, even when the unrestricted threshold in (iii) is exceeded.

If all real wavenumbers above the cutoff are possible, an equivalent strict threshold isIt is unchanged when $k_0^2\le \sqrt{A\gamma /D}$, and becomes $AD+\gamma +D{k}_0^2+A\gamma /k_0^2$ when $k_0^2>\sqrt{A\gamma /D}$. A bounded geometry generally gives a discrete spectrum, so the stronger requirement is that an actual allowed eigenvalue falls in the band.