In this section, we prove the main conclusion of this paper - the asymptotic behaviour of solution for problem (1.1)-(1.3).
Lemma 4.1 Let (H1) and (H2) hold, , . Then, for the approximate solutions of problem (1.1)-(1.3) constructed in the proof of Theorem 3.2, the following hold:
-
(i)
(4.1)
-
(ii)
Furthermore, if and , then for sufficiently large n, there exists a such that
(4.2)
Proof (i) Multiplying (3.1) by and summing for s, we get (4.1).
-
(ii)
From
it follows that there exists a such that
(4.3)
From (3.2), (3.3) and (4.3), it follows that for sufficiently large n. Hence from (3.5) we have
and
which gives
and
which together with for sufficiently large n gives
and
Hence, by Lemma 2.3, we have or . So, we have
□
Theorem 4.2 Let (H1) and (H2) hold, , . Assume that , . Then, for the global weak solution u given in Theorem 3.2, there exist positive constants C and λ such that
(4.4)
Proof Let be the approximate solutions of problem (1.1)-(1.3) in the proof of Theorem 3.2, then (3.4) holds. Multiplying (3.4) by (), we get
and
(4.5)
From (H2), Lemma 2.2 and Lemma 4.1, we get
where
Hence we have
and
From
we get
and
which together with for sufficiently large n gives
(4.8)
and
(4.9)
From (4.8) and the Poincaré inequality , it follows that there exists a constant such that
(4.10)
From (4.5)-(4.10) it follows that there exists a such that
Choose α such that
Then from (4.11) we get
From this and the Gronwall inequality, we get
and
(4.12)
On the other hand, from (4.8) we get
Hence, there exists a such that
(4.13)
Let be the subsequence of in the proof of Theorem 3.2. Then from (4.13) and(4.12), we obtain
which gives (4.4), where , . □