Research Article
Received 12 June 2014 Published online in Wiley Online Library
(wileyonlinelibrary.com) DOI: 10.1002/mma.3369
MOS subject classification: 35B40; 35B41; 37B55
Asymptotic behavior of solutions to a class of
nonlocal non-autonomous diffusion equations
F. D. M. Bezerra
a
*
†
, M. J. D. Nascimento
b
and S. H. da Silva
c
Communicated by A. Miranville
In this paper, we consider the nonlocal non-autonomous evolution problems
u
t
Da.t/u C g .ˇ.Ku// in ,
u D 0 in R
N
n,
where is a bounded smooth domain in R
N
, N 1, ˇ is a positive constant, the coefficient a is a continuous bounded
function on R, and K is an integral operator with symmetric kernel .Ku/.x/ :D
R
R
N
J.x, y/u.y/dy, being J a non-negative
function continuously differentiable on R
N
R
N
and
R
R
N
J., y/dy D 1. We prove the existence of global pullback attractor,
and we exhibit a functional to evolution process generated by this problem that decreases along of solutions. Assuming
the parameter ˇ is small enough, we show that the origin is locally pullback asymptotically stable. Copyright © 2014 John
Wiley & Sons, Ltd.
Keywords: pullback attractors; nonlocal diffusion equations; non-autonomous equations; evolution process
1. Introduction
The purpose of this paper is to prove existence of pullback attractor and existence of a functional, which decreases along solutions for
the nonlocal non-autonomous evolution equation in :
u
t
Da.t/u C g.ˇ.Ku//, (1.1)
where u D 0 in R
N
n, is a bounded open subset (not necessary smooth) of R
N
, and ˇ> 0. We will assume that a is a continuous
function in R and that there exists constants a
0
and a
1
such that
0 < a
0
a.t/ a
1
, for all t 2 R,
and K is an integral operator with symmetric kernel on L
2
.R
N
/, which is given by
.Ku/.t, x/ :D
Z
R
N
J.x, y/u.t, y/dy, for all t 2 R, (1.2)
where J is a non-negative function continuously differentiable on R
N
R
N
with bounded derivative,
R
R
N
J.x, y/ dy D 1 for all
x 2 R
N
, and
sup
x2R
N
Z
R
N
@
x
J.x, y/dy S and sup
y2R
N
Z
R
N
@
x
J.x, y/dx S,
for some constant 0 < S < 1.
a
Departamento de Matemática, Universidade Federal da Paraíba, 58051-900, João Pessoa - PB, Brazil
b
Departamento de Matemática, Universidade Federal de São Carlos, 13565-905 São Carlos - SP, Brazil
c
Unidade Acadêmica de Matemática, Universidade Federal de Campina Grande, 58051-900 Campina Grande - PB, Brazil
* Correspondence to: Flank David Morais Bezerra, Departamento de Matemática, Universidade Federal da Paraíba, 58051-900, João Pessoa - PB, Brazil.
†
E-mail: flank@mat.ufpb.br
Copyright © 2014 John Wiley & Sons, Ltd. Math. Meth. Appl. Sci. 2014