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.
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 .
Let be the weak closure of . Assume that , is weakly continuous, and
has a bounded uniformly (w.r.t. ) absorbing set ,
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 .
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 .
A function is tr.c. in if and only if
for any the set is precompact in ;
there exists a function , such that
Now, one introduces a class of function.
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 .
For every , ,
and . However, for every positive integer , and for any positive ,
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 .
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.
is a closed subspace of .
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.
Every translation compact function in is locally asymptotic smallness.
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
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 .
Assume that hold and is locally asymptotic smallness. Then for any , there exist positive integer and such that if , , satisfies
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
We now estimate the integral term on the right-hand side of (4.27).
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.
Assume that hold, let , . If in and weakly in , then for any ,
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.