This book, "Geometric Theory of Semilinear Parabolic Equations" by Daniel Henry, is a comprehensive treatment of the geometric theory of semilinear parabolic equations. It is part of the Lecture Notes in Mathematics series, edited by A. Dold and B. Eckmann. The book is structured into ten chapters, each covering different aspects of the theory. Chapter 1 provides an introduction and preliminary concepts, including definitions of geometric theory, basic facts, sectorial operators, and analytic semigroups. Chapter 2 presents various examples of nonlinear parabolic equations from physics, biology, and engineering. Chapter 3 discusses the existence, uniqueness, and continuous dependence of solutions. Chapter 4 covers dynamical systems and Liapunov stability. Chapter 5 examines the neighborhood of an equilibrium point, including stability, instability, and the Chafee-Infante problem. Chapter 6 explores invariant manifolds near an equilibrium point, including bifurcation and stability transfer. Chapter 7 deals with linear nonautonomous equations, including evolution operators, periodic systems, and exponential dichotomies. Chapter 8 focuses on the neighborhood of a periodic solution, including stability, orbital stability, and bifurcation. Chapter 9 discusses the neighborhood of an invariant manifold, including existence, stability, and coordinate systems. Chapter 10 provides two examples, including a selection-migration model and a problem in combustion theory. The book also includes notes, references, and an index. The book is a valuable resource for researchers and students in the field of partial differential equations and dynamical systems.This book, "Geometric Theory of Semilinear Parabolic Equations" by Daniel Henry, is a comprehensive treatment of the geometric theory of semilinear parabolic equations. It is part of the Lecture Notes in Mathematics series, edited by A. Dold and B. Eckmann. The book is structured into ten chapters, each covering different aspects of the theory. Chapter 1 provides an introduction and preliminary concepts, including definitions of geometric theory, basic facts, sectorial operators, and analytic semigroups. Chapter 2 presents various examples of nonlinear parabolic equations from physics, biology, and engineering. Chapter 3 discusses the existence, uniqueness, and continuous dependence of solutions. Chapter 4 covers dynamical systems and Liapunov stability. Chapter 5 examines the neighborhood of an equilibrium point, including stability, instability, and the Chafee-Infante problem. Chapter 6 explores invariant manifolds near an equilibrium point, including bifurcation and stability transfer. Chapter 7 deals with linear nonautonomous equations, including evolution operators, periodic systems, and exponential dichotomies. Chapter 8 focuses on the neighborhood of a periodic solution, including stability, orbital stability, and bifurcation. Chapter 9 discusses the neighborhood of an invariant manifold, including existence, stability, and coordinate systems. Chapter 10 provides two examples, including a selection-migration model and a problem in combustion theory. The book also includes notes, references, and an index. The book is a valuable resource for researchers and students in the field of partial differential equations and dynamical systems.