Differ Equ Dyn Syst (Jan&Apr 2011) 19(1&2):87–96
DOI 10.1007/s12591-010-0072-0
ORIGINAL RESEARCH
Existence and Upper Semicontinuity of Global Attractors
for Neural Network in a Bounded Domain
Severino Horácio da Silva
Published online: 4 January 2011
© Foundation for Scientific Research and Technological Innovation 2010
Abstract In this work we prove the existence of a compact global attractor for the flow of
the equation
∂ m(r, t )
∂ t
=-m(r, t ) + J ∗ ( f ◦ m)(r, t ) + h, h,> 0,
in L
2
( S
1
). We also give uniform estimates on the size of the attractor and show that the family
of attractors {A
J
} is upper semicontinuous at J
0
.
Keywords Well posedness · Global attractors · Upper semicontinuity
Mathematics Subject Classification (2000) 34G20 · 47H15
Introduction
We consider here the non local evolution equation
∂ u (x , t )
∂ t
=-u (x , t ) + J ∗ ( f ◦ u )(x , t ) + h, h > 0, (1)
where u (x , t ) is a real function on R × R
+
, h is a positive constant, J ∈ C
1
(R) is a non neg-
ative even function supported in the interval [-1, 1], and, f is a non negative nondecreasing
function. The ∗ above denotes convolution product, namely:
( J ∗ u )(x ) =
R
J (x - y )u ( y )dy .
Equation 1 was derived in 1972 by Wilson and Cowan [21], to model a single layer of
neurons. The function u (x , t ) denotes the mean membrane potential of a patch of tissue
S. H. da Silva (B )
Unidade Acadêmica de Matemática e Estatística UAME/CCT/UFCG, Bairro Universitário, CEP
58429-970, Rua Aprígio Veloso, 882, Campina Grande, PB, Brasil
e-mail: horacio@dme.ufcg.edu.br; horaciousp@gmail.com
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