= Dirichlet heat-reaction threshold on the unit square
{c}
{title2=$\kappa\leq2\pi^2$}
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 $\kappa\leq2\pi^2$ 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.
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