= All-time boundedness of a wave equation with a reaction term
{title2=$\alpha<\lambda_1$}
For $u_{tt}=\Delta u+\alpha u$ with a homogeneous <Dirichlet boundary condition>, let $\lambda_1>0$ be the first <Dirichlet Laplacian eigenvalue>. On a bounded one-dimensional interval, finite-energy displacement remains bounded for all time and all initial data exactly when $\alpha<\lambda_1$. The conserved energy $\tfrac12(\|u_t\|_2^2+\|\nabla u\|_2^2-\alpha\|u\|_2^2)$ controls the gradient and, by the <Poincare inequality> and one-dimensional integration, the maximum <norm>. At equality a mode can grow linearly; above it a mode can grow exponentially. Bounded oscillations need not have a long-time limit.
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