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.
Wei-Ming Ni holds a joint appointment as the Director of the Center for Partial Differential Equations at East China Normal University in Shanghai and as Professor of Mathematics at the University of Minnesota. His research interests include the qualitative properties of solutions to nonlinear elliptic and parabolic equations.
Preface; 1. Introduction: the heat equation; 2. Dynamics of general reaction-diffusion equations and systems; 3. Qualitative properties of steady states of reaction-diffusion equations and systems; 4. Diffusion in heterogeneous environments: 2 x 2 Lotka-Volterra competition systems; 5. Beyond diffusion: directed movements, taxis, and cross-diffusion; Bibliography; Index.