Nonlocal Diffusion Problems
About this Title
Fuensanta Andreu-Vaillo, José M. Mazón, Universitat de València, València, Spain, Julio D. Rossi, Universidad de Alicante, Alicante, Spain and J. Julián Toledo-Melero, Universitat de València, València, Spain
Publication: Mathematical Surveys and Monographs
Publication Year 2010: Volume 165
ISBNs: 978-0-8218-5230-9 (print); 978-1-4704-1392-7 (online)
MathSciNet review: MR2722295
MSC: Primary 35-02; Secondary 35K55, 35R09, 47H20, 76R50
Nonlocal diffusion problems arise in a wide variety of applications, including biology, image processing, particle systems, coagulation models, and mathematical finance. These types of problems are also of great interest for their purely mathematical content.
This book presents recent results on nonlocal evolution equations with different boundary conditions, starting with the linear theory and moving to nonlinear cases, including two nonlocal models for the evolution of sandpiles. Both existence and uniqueness of solutions are considered, as well as their asymptotic behaviour. Moreover, the authors present results concerning limits of solutions of the nonlocal equations as a rescaling parameter tends to zero. With these limit procedures the most frequently used diffusion models are recovered: the heat equation, the $p$-Laplacian evolution equation, the porous media equation, the total variation flow, a convection-diffusion equation and the local models for the evolution of sandpiles due to Aronsson-Evans-Wu and Prigozhin.
Readers are assumed to be familiar with the basic concepts and techniques of functional analysis and partial differential equations. The text is otherwise self-contained, with the exposition emphasizing an intuitive understanding and results given with full proofs. It is suitable for graduate students or researchers.
The authors cover a subject that has received a great deal of attention in recent years. The book is intended as a reference tool for a general audience in analysis and PDEs, including mathematicians, engineers, physicists, biologists, and others interested in nonlocal diffusion problems.
Graduate students and research mathematicians interested in diffusion problems and nonlinear PDEs.
Table of Contents
- 1. The Cauchy problem for linear nonlocal diffusion
- 2. The Dirichlet problem for linear nonlocal diffusion
- 3. The Neumann problem for linear nonlocal diffusion
- 4. A nonlocal convection diffusion problem
- 5. The Neumann problem for a nonlocal nonlinear diffusion equation
- 6. Nonlocal $p$-Laplacian evolution problems
- 7. The nonlocal total variation flow
- 8. Nonlocal models for sandpiles
- 9. Nonlinear semigroups