 Solution

Write $h=\Delta x=1/(M+1)$. The semidiscrete matrix is $B=h^{-2}\mathrm{tridiag}(1,-2,1)+\kappa I$. For $j=1,\dots ,M$, set $v_m^{(j)}=\sin (j\pi m/(M+1))$. The endpoint values vanish and the sine addition formula givesThese form an orthogonal basis of eigenvectors. Every initial vector decays precisely when every eigenvalue is negative. Since the largest is ${\lambda }_1$, the necessary and sufficient condition isNecessity follows by taking the first eigenvector as the initial data; at equality it remains unchanged. Sufficiency follows by decomposing any vector into the displayed decaying modes. The discrete reaction-diffusion spectral decay threshold is below ${\pi }^2$, since $\sin x<x$ for $x>0$, and equals ${\pi }^2-{\pi }^4{h}^2/12+O(h^4)$ as the mesh is refined. Therefore a coarse semidiscretization can spuriously grow for some parameters for which the continuous equation still decays.