We consider: the homogenization problem
\[
{
(
∂
u
ε
/
∂
t
)
(
x
,
t
)
+
β
ε
(
x
)
u
ε
(
x
,
t
)
=
0
,
a
m
p
;
t
⩽
0
,
u
ε
(
x
,
0
)
=
ϕ
(
x
)
,
\begin {cases} (\partial u\varepsilon /\partial t)(x,t) + \beta _\varepsilon (x) u_\varepsilon (x,t) = 0, & t\leqslant 0, \\ u_\varepsilon (x,0) = \phi (x), \end {cases}
\]
where
β
\beta
is a strictly positive bounded real function, periodic of period
1
1
, and
β
ε
(
x
)
=
β
(
x
/
ε
)
{\beta _\varepsilon }(x) = \beta (x/\varepsilon )
; the equivalent integral equation
\[
u
ε
(
x
,
t
)
+
∫
0
t
β
ε
(
x
)
u
ε
(
x
,
s
)
d
s
=
ϕ
(
x
)
;
{u_\varepsilon }(x,t) + \int _0^t {{\beta _\varepsilon }(x)\,{u_\varepsilon }(x,s)\;ds = \phi (x)};
\]
and the homogenized equation
\[
u
0
(
x
,
t
)
+
∫
0
t
K
(
t
−
s
)
u
0
(
s
)
d
s
=
ϕ
(
x
)
,
{u_0}(x,t) + \int _0^t {K(t - s)\,{u_0}(s)\,ds = \phi (x)},
\]
where
K
K
is a unique, well-defined function depending on
β
\beta
. We study this problem for a time dependent
β
\beta
, and characterize a two-variable function
K
(
s
,
t
)
K(s,t)
satisfying
\[
u
0
(
x
,
t
)
+
∫
0
t
K
(
s
,
t
−
s
)
u
0
(
x
,
s
)
d
s
=
ϕ
(
x
)
{u_0}(x,t) + \int _0^t {K(s,t - s)\,{u_0}(x,s)\;ds = \phi (x)}
\]
and study its uniqueness.