Dirichlet heat-reaction threshold on the unit square ($\kappa \le 2{\pi }^2$)

For zero Dirichlet boundary conditions on the unit square, the modes $2\sin (j\pi x)\sin (k\pi y)$ have heat-reaction growth rates $\kappa -{\pi }^2(j^2+k^2)$. Their largest rate is $\kappa -2{\pi }^2$. Hence the solution is an L2 norm contraction for $\kappa \le 2{\pi }^2$ and decays uniformly at positive times as $t\to \infty$ when the inequality is strict. At equality the first sine mode is stationary. For larger fixed $\kappa$ the equation remains well posed on every finite time interval, but has a growing mode; long-time contractivity is a different property from finite-time well-posedness.