Julio D. Rossi

Nonlocal Diffusion Problems

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

Non-simulataneous blow-up in a semilinear parabolic system

p

Isolation and simplicity for the first eigenvalue of the p-Laplacian with a nonlinear boundary condition

-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.

A convex–concave problem with a nonlinear boundary condition

Abstract. In this paper we study the possibility of non-simultaneous blow-up for positive solutions of a fully coupled system of two semilinear heat equations,

A NONLOCAL p-LAPLACIAN EVOLUTION EQUATION WITH NONHOMOGENEOUS DIRICHLET BOUNDARY CONDITIONS

u_t=\Delta u + u^{p_{11}} v^{p_{12}}, v_t=\Delta v + u^{p_{21}} v^{p_{22}}

An Asymptotic Mean Value Characterization for a Class of Nonlinear Parabolic Equations Related to Tug-of-War Games

. Under adequate hypotheses, we prove that if u blows up then v can fail to blow up if and only if

The Blow-up Rate for a System of Heat Equations with Non-trivial Coupling at the Boundary

p_{11} >1

Porous medium equation with absorption and a nonlinear boundary condition

and

An adaptive time step procedure for a parabolic problem with blow-up

p_{21} < p_{11}-1

Critical exponents for a semilinear parabolic equation with variable reaction

.