Where do solutions go, and how do they behave en route? These are two of the major questions addressed by the qualita- tive theory of differential equations. The purpose of this book is to answer these questions for certain classes of equa- tions by recourse to the framework of semidynamical systems (or topological dynamics as it is sometimes called). This approach makes it possible to treat a seemingly broad range of equations from nonautonomous ordinary differential equa- tions and partial differential equations to stochastic differ- ential equations. The methods are not limited to the examples presented here, though. The basic idea is this: Embed some representation of the solutions of the equation (and perhaps the equation itself) in an appropriate function space. This space serves as the phase space for the semidynamical system. The phase map must be chosen so as to generate solutions to the equation from an initial value. In most instances it is necessary to provide a "weak" topology on the phase space. Typically the space is infinite dimensional. These considerations motivate the requirement to study semidynamical systems in non locally compact spaces. Our objective here is to present only those results needed for the kinds of applications one is likely to encounter in differen- tial equations. Additional properties and extensions of ab- stract semidynamical systems are left as exercises. The power of the semidynamical framework makes it possible to character- Preface ize the asymptotic behavior of the solutions of such a wide class of equations.
Semidynamical Systems in Infinite Dimensional Spaces
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