**Mathematical Surveys and Monographs**

Volume: 165;
2010;
256 pp;
Hardcover

MSC: Primary 45; 47; 35;

Print ISBN: 978-0-8218-5230-9

Product Code: SURV/165

List Price: $87.00

Individual Member Price: $69.60

**Electronic ISBN: 978-1-4704-1392-7
Product Code: SURV/165.E**

List Price: $87.00

Individual Member Price: $69.60

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#### Supplemental Materials

# Nonlocal Diffusion Problems

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*Fuensanta Andreu-Vaillo; José M. Mazón; Julio D. Rossi; J. Julián Toledo-Melero*

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.

This book is published in cooperation with Real Sociedad Matemática Española (RSME)

#### Readership

Graduate students and research mathematicians interested in diffusion problems and nonlinear PDEs.

#### Reviews & Endorsements

The results of this book are given with complete proofs and also an emphasis on the intuitive understanding of the results. This extends the audience beyond mathematicians to include engineers, physicists and biologists with a good background in Analysis and PDEs.

-- Mathematical Reviews

#### Table of Contents

# Table of Contents

## Nonlocal Diffusion Problems

- Contents vii8 free
- Preface xi12 free
- Chapter 1. The Cauchy problem for linear nonlocal diffusion 118 free
- Chapter 2. The Dirichlet problem for linear nonlocal diffusion 3148
- Chapter 3. The Neumann problem for linear nonlocal diffusion 4158
- Chapter 4. A nonlocal convection diffusion problem 6582
- Chapter 5. The Neumann problem for a nonlocal nonlinear diffusion equation 99116
- Chapter 6. Nonlocal $p$-Laplacian evolution problems 123140
- Chapter 7. The nonlocal total variation flow 163180
- Chapter 8. Nonlocal models for sandpiles 191208
- Appendix A. Nonlinear semigroups 223240
- Bibliography 249266
- Index 255272 free