= All-time boundedness of a semidiscrete reaction wave equation
{title2=$\alpha<\lambda_{\min}(K_h)$}
For $U''=(\alpha I-K_h)U$ 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.
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