A Hilbert space approach to instability in semilinear partial differential equations

A method of R. Bellman in the instability theory of autonomous nonlinear ODE is enlarged to incompass semilinear evolution equations of the form u￠+Lu=f(u)u'+Lu=f(u) in the Hilbert space framework. As a consequence it is shown that linearized instability of solutions to parabolic problems with linear damping in a bounded domain implies the Liapounov instability in those topologies for which the question is meaningful.