The Mathematics of Diffusion

Diffusion has been used extensively in many scientific disciplines to model a wide variety of phenomena. The Mathematics of Diffusion focuses on the qualitative properties of solutions to nonlinear elliptic and parabolic equations and systems in connection with domain geometry, various boundary conditions, the mechanism of different diffusion rates, and the interaction between diffusion and spatial heterogeneity. The book systematically explores the interplay between different diffusion rates from the viewpoint of pattern formation, particularly Turing's diffusion driven instability in both homogeneous and heterogeneous environments, and the roles of random diffusion, directed movements, and spatial heterogeneity in the classical Lotka-Volterra competition systems. Interspersed throughout the book are many simple, fundamental, and important open problems for readers to investigate. Audience: This book is intended for researchers and graduate students in the areas of elliptic or parabolic equations and in mathematical biology. Contents: Preface; Chapter 1. Introduction: The Heat Equation; Chapter 2. Dynamics of General Reaction-Diffusion Equations and Systems; Chapter 3. Qualitative Properties of Steady States of Reaction-Diffusion Equations and Systems; Chapter 4. Diffusion in Heterogeneous Environments: 2 x 2 Lotka-Volterra Competition Systems; Chapter 5. Beyond Diffusion: Directed Movements, Taxis, and Cross-Diffusion; Bibliography; Index.