Linear and non-linear semigroups and applications

The theory of semigroups of operators is undoubtedly one of the most powerful and elegant tools in modern functional analysis for the study of evolution equations. From the classical heat diffusion to complex wave propagation phenomena and quantum mechanics, the abstract language of semigroups allows us to unify diverse problems under a common framework, providing robust methods for establishing existence, uniqueness, and asymptotic behavior of solutions. This book, Linear and Nonlinear Semigroups and Applications, is the result of years of teaching and research at the State University of Maringá. It has been conceived to serve both as a textbook for graduate students in Mathematics and as a reference for researchers interested in the analysis of partial differential equations. The text is structured to guide the reader from the foundations to the frontiers of the theory. We begin with a review of differential and integral calculus in Banach spaces, setting the stage for the theory of C0-semigroups of linear operators. Here, the classical theorems of Hille-Yosida and Lumer-Phillips are presented not just as abstract results, but as operational tools essential for solving linear Evolution problems. However, nature is inherently nonlinear. A distinctive feature of this volume is the substantial treatment dedicated to nonlinear analysis. We introduce the theory of monotone and accretive operators, multivalued mappings, and the crucial Crandall-Liggett Theorem, which generalizes the generation of semigroups to the nonlinear setting. This transition is handled with care, highlighting the geometric and analytic subtleties that arise when linearity is abandoned. Throughout the book, the abstract theory is constantly motivated by and applied to concrete problems. We explore in detail the heat equation, the wave equation with various types of damping (frictional, viscoelastic, and boundary damping), and the Schrödinger equation. Special attention is given to the regularity of solutions and to the concept of weak and generalized solutions, bridging the gap between abstract functional analysis and applied mathematics. We assume the reader has a background in basic functional analysis and Lebesgue integration theory. Our goal is that, by the end of this journey, the reader will not only understand the "how" and "why" of semigroup theory but will also be equipped to apply these powerful techniques to their own research problems. We are grateful to our colleagues and students whose questions and feedback over the years have helped shape this material. We hope this book serves as a solid foundation for those venturing into the vast and dynamic field of evolution equations.