This book provides an introduction to semilinear evolution equations, focusing on the theory and applications of these equations in various contexts. The authors, Thierry Cazenave and Alain Haraux, present a comprehensive overview of the subject, covering both theoretical foundations and practical examples. The book is structured into seven chapters, each addressing different aspects of semilinear evolution equations. The first chapter introduces preliminary results, including abstract tools, the exponential of a linear operator, Sobolev spaces, and vector-valued functions. The second chapter discusses m-dissipative operators, their properties, and applications in Banach and Hilbert spaces. The third chapter presents the Hille-Yosida-Phillips theorem and its applications, including the generation of semigroups and their properties. The fourth chapter addresses inhomogeneous equations and abstract semilinear problems, including local and global existence, continuous dependence on initial data, and regularity. The fifth chapter focuses on the heat equation, discussing local and global existence, blow-up in finite time, and applications to a model case. The sixth chapter examines the Klein-Gordon equation, covering local and global existence, blow-up in finite time, and applications. The seventh chapter explores the Schrödinger equation, including linear and nonlinear cases, local and global existence, and blow-up in finite time. The book is a valuable resource for researchers and students in the field of partial differential equations, offering a thorough treatment of semilinear evolution equations and their applications.This book provides an introduction to semilinear evolution equations, focusing on the theory and applications of these equations in various contexts. The authors, Thierry Cazenave and Alain Haraux, present a comprehensive overview of the subject, covering both theoretical foundations and practical examples. The book is structured into seven chapters, each addressing different aspects of semilinear evolution equations. The first chapter introduces preliminary results, including abstract tools, the exponential of a linear operator, Sobolev spaces, and vector-valued functions. The second chapter discusses m-dissipative operators, their properties, and applications in Banach and Hilbert spaces. The third chapter presents the Hille-Yosida-Phillips theorem and its applications, including the generation of semigroups and their properties. The fourth chapter addresses inhomogeneous equations and abstract semilinear problems, including local and global existence, continuous dependence on initial data, and regularity. The fifth chapter focuses on the heat equation, discussing local and global existence, blow-up in finite time, and applications to a model case. The sixth chapter examines the Klein-Gordon equation, covering local and global existence, blow-up in finite time, and applications. The seventh chapter explores the Schrödinger equation, including linear and nonlinear cases, local and global existence, and blow-up in finite time. The book is a valuable resource for researchers and students in the field of partial differential equations, offering a thorough treatment of semilinear evolution equations and their applications.