Reaction-diffusion spectral decay threshold ($\kappa <{\pi }^2,{\kappa }_h=4h^{-2}{\sin }^2(\pi h/2)$)

On the unit interval with zero Dirichlet boundary conditions, $u_t=u_{xx}+\kappa u$ has sine-mode growth rates $\kappa -j^2{\pi }^2$. Every finite-energy initial condition decays exactly when the largest rate is negative, namely $\kappa <{\pi }^2$. A centered finite difference grid with $h=1/(M+1)$ has eigenvectors $\sin (j\pi m/(M+1))$ and growth rates $\kappa -4h^{-2}{\sin }^2(j\pi /(2(M+1)))$. Its all-data decay threshold is the displayed ${\kappa }_h$, strictly less than ${\pi }^2$ and approaching it from below. Equality leaves a stationary lowest mode; exceeding the threshold allows growth.