Preface

The goal in this monograph is to present recent results concerning nonlocal

evolution equations with different boundary conditions. We deal with existence and

uniqueness of solutions and their asymptotic behaviour. We also give some results

concerning limits of solutions to nonlocal equations when a rescaling parameter

goes to zero. We recover in these limits some of the most frequently used diffusion

models such as the heat equation, the p-Laplacian evolution equation, the porous

medium equation, the total variation flow and a convection-diffusion equation. This

book is based mainly on results from the papers [14], [15], [16], [17], [68], [78],

[79], [80], [120], [121] and [140].

First, let us briefly introduce the prototype of nonlocal problems that will be

considered in this monograph. Let J : RN → R be a nonnegative, radial, continuous

function with

RN

J(z) dz = 1.

Nonlocal evolution equations of the form

(0.1) ut(x, t) = (J ∗ u − u)(x, t) =

RN

J(x − y)u(y, t) dy − u(x, t),

and variations of it, have been recently widely used to model diffusion processes.

More precisely, as stated in [106], if u(x, t) is thought of as a density at a point

x at time t and J(x − y) is thought of as the probability distribution of jumping

from location y to location x, then

RN

J(y − x)u(y, t) dy = (J ∗ u)(x, t) is the rate

at which individuals are arriving at position x from all other places and −u(x, t) =

−

RN

J(y − x)u(x, t) dy is the rate at which they are leaving location x to travel

to all other sites. This consideration, in the absence of external or internal sources,

leads immediately to the fact that the density u satisfies equation (0.1).

Equation (0.1) is called nonlocal diffusion equation since the diffusion of the

density u at a point x and time t depends not only on u(x, t) and its derivatives, but

also on all the values of u in a neighborhood of x through the convolution term J ∗u.

This equation shares many properties with the classical heat equation, ut = Δu,

such as: bounded stationary solutions are constant, a maximum principle holds for

both of them and, even if J is compactly supported, perturbations propagate with

infinite speed, [106]. However, there is no regularizing effect in general.

Let us fix a bounded domain Ω in

RN

. For local problems the two most common

boundary conditions are Neumann’s and Dirichlet’s. When looking at boundary

conditions for nonlocal problems, one has to modify the usual formulations for local

problems. As an analog for nonlocal problems of Neumann boundary conditions

xi