SEMIGROUP THEORY AND EVOLUTION EQUATIONS

Abstract

The present dissertation provides information about the fundamental concepts of Semigroup Theory and its applications to differential equations (Cauchy problem). In this project, the basis for our study is the evolution equation. When we solve a Cauchy problem we get a family of evolution operators that can take the starting condition of a system and predict its state at a later time that means solving the Cauchy problem is the same as finding a family of evolution operators E(t) and the properties which operator E(t) satisfies are called the semigroup properties. The study begins with a comprehensive review of the basic definitions and properties of semigroups, the project delves into the classification of semigroups based on various structural properties and algebraic operations. Here special attention is given to the classification of semigroups such as Uniformly continuous semigroups, Strongly continuous semigroups, Contraction semigroups, and some key concepts of these semigroups like generator, resolvent, infinitesimal generators, etc. Here we study the Hille-Yosida Theorem and Lumer-Phillips Theorem which emerge as a powerful tool for understanding the structure and complexity of Contraction semigroup. We also study the application of semigroup in various areas of PDEs such as Heat equation, transport equation, etc. The dissertation concludes with reflections on semigroups, their properties, and applications.