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
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