Stability and Hopf

We develop the principle of linearized stability and a Hopf bifurcation theorem as elements of a geometric theory for fully nonlinear parabolic-hyperbolic problems. Crucial steps in our work are showing the diierentiability of the time-t map, showing that the admissible initial data form a manifold (whose failure to be linear is due to the general boundary conditions we study), and analyzing the spectrum of the generator of the linearized semigroup. This paper provides the abstract framework for the study of a class of concrete problems of self-sustained oscillations of nonlinearly viscoelastic bodies as in Antman and Koch 7]. Our equations are intrinsically interesting: They provide an example of a new kind of semiiow that combines properties of ordinary diierential equations and parabolic equations in a novel way.