A Hilbert space approach to instability in semilinear partial differential equations (Q5946715)

The method of R. Bellman in the instability theory of autonomous nonlinear ordinary differential equations is generalized to semilinear evolution equations in a Hilbert space of the form \(u' + Lu = f(u)\) where \(L\) is a linear selfadjoint operator and \(f\) is a continuous nonlinearity. As a consequence, it is shown that the linearized instability of solutions to parabolic equations with linear damping in a bounded domain implies the Lyapunov instability. The instabiliy of hyperbolic equations is also discussed.