Dirichlet reaction-diffusion growth threshold in a cuboid ($s_{111}=\gamma -\mu {\pi }^2(L_x^{-2}+L_y^{-2}+L_z^{-2})$)

For $n_t=\mu \Delta n+\gamma n$ in a rectangular box with zero Dirichlet boundary conditions, separation of variables gives spatial eigenfunctions $\sin (p\pi x/L_x)\sin (q\pi y/L_y)\sin (r\pi z/L_z)$ and temporal rates $\gamma -\mu {\pi }^2(p^2/L_x^2+q^2/L_y^2+r^2/L_z^2)$. The largest rate is the displayed one. Every nonzero nonnegative initial density has positive projection onto this first eigenfunction, so its sign determines eventual exponential growth or decay; equality leaves a stationary first mode. At fixed volume, write $a_j=L_j^{-2}$: their product is fixed, and the arithmetic-geometric mean inequality minimizes their sum when all three are equal. Thus the cube maximizes the growth rate among rectangular boxes of fixed volume; making a box sufficiently thin suppresses growth.