An Introduction to Semiflows (Chapman and Hall /Crc Monographs and Surveys in Pure and Applied Mathematics)

Semiflows are a class of Dynamical Systems, meaning that they help to describe how one state develops into another state over the course of time, a very useful concept in Mathematical Physics and Analytical Engineering. The authors concentrate on surveying existing research in continuous semi-dynamical systems, in which a smooth action of a real number on another object occurs from time zero, and the book proceeds from a grounding in ODEs through Attractors to Inertial Manifolds. The book demonstrates how the basic theory of dynamical systems can be naturally extended and applied to study the asymptotic behavior of solutions of differential evolution equations.

This book introduces the class of dynamical systems called semiflows, which includes systems defined or modeled by certain types of differential evolution equations (DEEs). It focuses on the basic results of the theory of dynamical systems that can be extended naturally and applied to study the asymptotic behavior of the solutions of DEEs. The authors concentrate on three types of absorbing sets: attractors, exponential attractors, and inertial manifolds. They present the fundamental properties of these sets, and then proceed to show the existence of some of these sets for a number of dynamical systems generated by well-known physical models. In particular, they consider in full detail two particular PDEEs: a semilinear version of the heat equation and a corresponding version of the dissipative wave equation. These examples illustrate the most important features of the theory of semiflows and provide a sort of template that can be applied to the analysis of other models.

The material builds in a careful, gradual progression, developing the background needed by newcomers to the field, and culminating in a more detailed presentation of the main topics than found in most sources. The authors' approach to and treatment of the subject builds the foundation for more advanced references and research on global attractors, exponential attractors, and inertial manifolds.

DYNAMICAL PROCESSESIntroductionOrdinary Differential EquationsAttracting SetsIterated SequencesLorenz' EquationsDuffing's EquationSummaryATTRACTORS OF SEMIFLOWSDistance and SemidistanceDiscrete and Continuous SemiflowsInvariant SetsAttractorsDissipativityAbsorbing Sets and AttractorsAttractors via a-ContractionsFractal DimensionA Priori EstimatesATTRACTORS FOR SEMILINEAR EVOLUTION EQUATIONSPDEEs as Dynamical SystemsFunctional FrameworkThe Parabolic ProblemThe Hyperbolic ProblemRegularityUpper Semicontinuity of the Global AttractorsEXPONENTIAL ATTRACTORSIntroductionThe Discrete Squeezing Proper "This book introduces the class of dynamical systems called semiflows, which includes systems defined or modeled by certain types of differential evolution equations (DEEs). It focuses on the basic results of the theory of dynamical systems that can be extended naturally and applied to study the asymptotic behavior of the solutions of DEEs."--Jacket This book provides an accessible introduction to the class of dynamical systems called semiflows, which includes systems defined or modeled by certain types of differential evolution equations (DEEs). Proceeding from a grounding in ordinary differential equations to attractors and inertial manifolds, the authors show how the basic theory of dynamic Semiflows are a class of dynamical systems, meaning that they help to describe how one state develops into another state over the course of time. This work concentrates on surveying existing research in continuous semi-dynamical systems In this chapter we introduce the definition of DYNAMICAL PROCESS, and the main ideas of the theory of dynamical systems that we want to investigate.