DISSIPATIVE SYSTEMS

Abstract

This chapter discusses dissipative systems. In relation to applications, one of the problems, as in the use of this lemma, in developing a general theory is to have results that depend upon determining the boundedness of motions. One cannot give direct tests for compactness but can verify boundedness by use. In this generality, one cannot expect solutions to be smoother than the initial data and further restrictions need to be placed. Certain types of parabolic and hyperbolic partial differential equations are known to define processes on appropriate Sobolev spaces. In the hyperbolic case, there is some smoothing of initial data but this is not so for hyperbolic equations. Thus, for ordinary differential equations, one know for dissipative ordinary differential equations, there is always a periodic solution of period ω and for dissipative retarded functional differential equations when the solution map maps bounded sets into bounded sets and ω ≥ r, there is a periodic solution of period ω.