 All-time boundedness of a semidiscrete reaction wave equation

For $U^{\prime \prime }=(\alpha I-K_h)U$ with symmetric positive definite $K_h$, all-data displacement is bounded for all time on a fixed grid exactly when $\alpha <{\lambda }_{\min }(K_h)$. Diagonalization of a matrix reduces this assertion to oscillators. At equality, an initial velocity in a zero-frequency mode produces linear growth; above it a growing hyperbolic mode is available. A lower bound on ${\lambda }_{\min }(K_h)-\alpha$ uniform in the mesh gives a uniform displacement estimate. This all-time statement is stronger than displacement stability of a symmetric semidiscrete wave equation on fixed finite time intervals.