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Take $M$ interior points in each direction and spacing $h=1/(M+1)$. The one-dimensional Dirichlet discrete Laplacian has sine eigenvectors $v_i^{(j)}=\sin (j\pi ih)$, becauseThe five-point Dirichlet Laplacian as a Kronecker sum therefore has product eigenvectors $v_i^{(j)}{v}_r^{(\ell )}$. Adding the reaction term gives modal growth ratesThe semidiscrete matrix is real symmetric, so these modes are orthogonal in the mesh-weighted discrete L2 norm. Non-growth for all times is equivalent to every rate being nonpositive. The largest is ${\lambda }_{11}$, giving the five-point heat-reaction stability thresholdAt equality the first grid mode is neutral; above it that mode grows exponentially. This criterion concerns the spatially semidiscrete, continuous-time evolution. Any subsequent time integrator has an additional stability requirement.

The threshold expands as $2{\pi }^2-{\pi }^4{h}^2/6+O(h^4)$ and is below the continuum threshold for every finite mesh. Thus at the continuum neutral value the grid first mode has a small positive $O(h^2)$ rate. This is compatible with finite-time convergence, although it prevents an all-time contraction on that fixed mesh. In the weaker finite-time mesh-uniform sense, all fixed $\kappa$ have the bound $\Vert U(t){\Vert }_h\le e^{\kappa t}\Vert U(0){\Vert }_h$ because the discrete Laplacian is nonpositive. The boxed range is the natural non-growing stability range, matching the threshold comparison in the preceding part; finite-time stability versus power boundedness is a distinct question.