CHAPTER 1

The Cauchy problem

for linear nonlocal diffusion

The aim of this chapter is to begin the study of the nonlocal evolution problems

by the analysis of the asymptotic behaviour of solutions of nonlocal linear diffusion

problems in the whole RN . First, we deal with the simplest model,

ut(x, t) =

RN

J(x − y)u(y, t) dy − u(x, t),

and after that we also treat a nonlocal analog of higher order problems,

ut(x, t) =

(−1)n−1

(J ∗ Id −

1)n

(u(x, t)).

We focus our attention on existence and uniqueness of solutions, their asymptotic

behaviour as t → ∞ and the convergence of solutions of these nonlocal evolution

equations to solutions of classical models, such as the heat equation, when the

nonlocal equation is rescaled in an appropriate way. As it happens in the study

of the Cauchy problem for the heat equation, the Fourier transform will play a

fundamental role, allowing us to obtain an explicit formula for the solution to the

nonlocal equation in Fourier variables.

1.1. The Cauchy problem

We consider the linear nonlocal diffusion problem presented in the Preface,

(1.1)

⎧

⎨

⎩

ut(x, t) = J ∗ u(x, t) − u(x, t) =

RN

J(x − y)u(y, t) dy − u(x, t),

u(x, 0) = u0(x),

for x ∈ RN and t 0. Here J satisfies the following hypothesis, which will be

assumed throughout this chapter:

(H) J ∈

C(RN

, R) is a nonnegative radial function with J(0) 0 and

RN

J(x) dx = 1.

This means that J is a radial probability density.

As we have mentioned in the Preface, this equation has been used to model

diffusion processes. More precisely (see [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

1

http://dx.doi.org/10.1090/surv/165/01