In this section, we establish the existence of uniform attractor for the non-autonomous lattice systems (2.8)–(2.10). Let
be a Banach space, and let
be a subset of some Banach space.
Definition 4.1.
is said to be
weakly continuous, if for any
, the mapping
is weakly continuous from
to
.
A family of processes
,
is said to be uniformly
-limit compact if for any
and bounded set
, the set
is bounded for every
and
is precompact set as
. We need the following result in [14].
Theorem 4.2.
Let
be the weak closure of
. Assume that
,
is
weakly continuous, and
-
(i)
has a bounded uniformly (w.r.t.
) absorbing set
,
-
(ii)
is uniformly (w.r.t.
)
-limit compact.
Then the families of processes
,
,
possess, respectively, compact uniform (w.r.t.
,
, resp.) attractors
and
satisfying
Furthermore,
is nonempty for all
.
Let
be a Banach space and
, denote the space
of functions
,
with values in
that are locally
-power integral in the Bochner sense, it is equipped with the local
-power mean convergence topology. Recall the Propositions in [12].
Proposition 4.3.
A set
is precompact in
if and only if the set
is precompact in
for every segment
. Here,
denotes the restriction of the set
to the segment
.
Proposition 4.4.
A function
is tr.c. in
if and only if
-
(i)
for any
the set
is precompact in
;
-
(ii)
there exists a function
,
such that
Now, one introduces a class of function.
Definition 4.5.
A function
is said to be locally asymptotic smallness if for any
, there exists positive integer
such that
Denote by
the set of all locally asymptotic smallness functions in
. It is easy to see that
. The next examples show that there exist functions in
but not in
, and a function belongs to
is not necessary a tr.c. function in
.
Example 4.6.
Let
,
For every
,
,
Thus,
and
. However, for every positive integer
, and for any positive
,
Therefore,
.
Example 4.7.
,
for
,
Here,
denote the positive integer set.
For every positive integer
,
, and for
,
which implies that
Therefore,
. Note that for any
,
From Proposition 4.4,
is not translation compact in
.
Remark 4.8.
Example 4.7 shows that a locally asymptotic function is not necessary translation compact in
.
In the following, we give some properties of locally asymptotic smallness function.
Lemma 4.9.
is a closed subspace of
.
Proof.
Let
such that
Then, for any
, there exists positive integer
such that for every
,
Since
, there exist
such that for all
,
Let
, we get that
Therefore,
. This completes the proof.
Lemma 4.10.
Every translation compact function
in
is locally asymptotic smallness.
Proof.
Since
is tr.c. in
, we get that
is precompact in
. By Proposition 4.3, we get that
is precompact in
. Thus, for any
, there exists finite number
such that for every
, there exist some
,
, such that
For the
given above,
implies that there exists positive integer
such that
Therefore,
which implies
is locally asymptotic smallness. This completes the proof.
We now establish the uniform estimates on the tails of solutions of (2.8)–(2.10) as
.
Lemma 4.11.
Assume that
hold and
is locally asymptotic smallness. Then for any
, there exist positive integer
and
such that if
,
,
satisfies
Proof.
Choose a smooth function
such that
for
, and
and there exists a constant
such that
for
. Let
be a suitable large positive integer,
. Taking the inner product of (2.8) with
and (2.9) with
in
, we have
From
, we have
where
is same as in Lemma 3.1
as in (3.10)
Summing up (4.22), from (4.23)–(4.25) we get
Thus,
We now estimate the integral term on the right-hand side of (4.27).
Similarly,
Since
is locally asymptotic smallness, from (4.27)–(4.29) we get that for any
, if
, there exist
and sufficient large positive integer
such that
The proof is completed.
Lemma 4.12.
Assume that
hold, let
,
. If
in
and
weakly in
, then for any
,
Proof.
Let
,
. Since
is bounded in
, by Lemma 3.2, we get that
Therefore, for all
,
,
Note that
is the solution of (2.8) and (2.9) with time symbol
, it follow from (4.32) that
In the following, we show that
. By the fact that
is the solution of (2.8) and (2.9), for any
, we get that
Note that
weakly in
. Let
in (4.35), by (4.34) we get that
is the solution of (2.8) and (2.9) with the initial data
. By the unique solvability of problem (2.8)–(2.10), we get that
. This completes the proof.
Proof of Theorem A.
From Lemmas 3.2, 4.11 and 4.12, and Theorem 4.2, we get the results.