The past forty years have witnessed an intensive study of problems in mathematical physics governed by dissipative equations and much progress has been achieved. Two already existing branches of mathematics have played a central role in these investigations: first, the qualitative theory of ordinary differential equations and - closely related to that - the theory of dynamical systems; second, the theory of partial differential equations. Thus, in this same connection, mathematicians have successfully applied finite dimensional concepts and techniques, suitably modified, to the study of semigroups generated in infinite dimensional spaces by evolutionary partial differential equations.
In recent years several authors have developed and exploited this combination of finite dimensional and infinite dimensional techniques. Specifically, we cite the monographs by J. K. Hale [HA 2], R. Temam [TE 1], A. V. Babin and M. I. Vishik [B-V 2], and O. A. Ladyzhenskaya [LA 3]. This present book is in that same vein. In it we shall set forth the theory of asymptotic behavior for dynamical systems corresponding to parabolic equations and - in that connection - we will expound the theory of global attractors.
An important notion in these developments is that of sectorial operator, an idea studied by A. Friedman [FR 1] and extensively exploited by D. Henry [HE 1] More recently, H. Amann [AM 5], A. Lunardi [LU 1] and H. Tanabe [TA 2] have employed sectorial operators in their own investigations, which have emphasized the union of finite dimensional and infinite dimensional methods.
Of course, a major theme in this present book will be the use of sectorial operators in the study of parabolic problems.