Global Attractors for the Higher-Order Evolution Equation

In this paper, we obtain the existence of a global attractor for the higher-order evolution type equation. Moreover, we discuss the asymptotic behavior of global solution.


Introduction
We consider the following nonlinear evolution equation where in a bounded domain Ω ⊂ R n with smooth boundary ∂ Ω, the assumption on f , g, u 0 and u 1 will be made below. When m = 1, the equation (1.1) is following form Chen and Wang [19] proved the existence of global attractor for the problem (1.2). Lately, Xie and Zhong in [8] studied the existence of global attractor of solution for the problem (1.1) with f = 0. Also, there are some authors studied the existence and nonexistence, asymptotic behavior of global solution for (1.2) (see [2][3][4][5][6][7] for more details ). Nakao and Yang in [9] showed the global attractor of the Kirchhoff type wave equation.
In this paper, we improve our result by adopting and modifying the method of [19], we studied more general form of the equation. This paper is organized as follows: In section 2, we give some assumptions and state the main results. In section 3, we prove the global existence of solution using the Faedo-Galerkin method. Also, we write some important estimates for the solution. In section 4, the existence of the global attractor is proved. In Section 5, the proof of decay property for solution is showed.

Preliminaries and main results
We write the Sobolev space H k (Ω) = W k,2 (Ω) , H k 0 (Ω) = W k,2 0 (Ω). Furthermore, we show by (., .) the inner product of L 2 (Ω) , by . p the norm of L p (Ω) , p ≥ 1 and by . E the norm of any other Banach space E. As usual, we give u (t) instead of u (x,t) , and u (t) for u t (t) and so on.
We write the following assumptions on f and g.
(2) {S (t)} t≥0 as asymptotically compact in E, that is, for any bounded sequence {y n } in E and t n → ∞ as n → ∞, {S (t n ) y n } ∞ n=1 has a convergent subsequence relating to E topology.
We show the basic results now.
In this theorem Show the solution in Theorem 1 by S (t) (u 0 , u 1 ) = (u (t) , u t (t)) . We are now in a position to prove some continuity of S (t) relating to the initial data (u 0 , u 1 ) , which will be needed for the proof of the existence of global attractor.
Theorem 4 denotes that the semigroup S (t) : X → X is continuous on X. For the decay property of solution u (t) for the problem (1.1), we get Theorem 6. Suppose u is a weak solution in Theorem 3 with f = 0 and g (x, u) = g (u) . Besides, suppose 0 ≤ 2G (u) ≤ ug (u) . Then, for whichever q > 0, there is C 1 = C 1 P The Proof of Theorem 3 In this section, we suppose that all assumptions in Theorem 3 are supplied. Firstly, we establish the global existence of a solution to problem (1.1) with Fadeo-Galerkin method as in [16,17].
Assume ω j (x) ( j = 1, 2, ...) is the complete set of properly normalized eigenfunctions for the operator (−∆) m in H m 0 (Ω). Then, the family {ω 1 , ω 2 ..., ω k , ...} holds an orthogonal basis for both H m 0 (Ω) and L 2 (Ω), see [16,17]. For each positive integer k, show V k = span {ω 1 , ω 2 ..., ω k , ...} . We search for an approximation solution u k (t) to the problem (1.1) in the form where d jk (t) are the solution of the nonlinear ordinary differential equation (ODE) system in the variant t: with the initial conditions where u 0k and u 1k are chosen in V k so that Here (., .) shows the inner product in L 2 (Ω) . Then, Sobolev imbedding theorem means that ∃c 0 > 0, such that and (3.1) shows that for any v ∈ V k , We know, the system (3.1) and (3.2) accept a unique solution u k (t) on the interval [0, T ] for any T > 0. Such a solution can be expanded to the overall interval [0, ∞). We show by C i (i = 1, 2, ...) the constants that are independent of k and t ≥ 0, by C 0 the constant depending on k 1 , k 2 in (A 2 ) and Sobolev imbedding constant c 0 in (3.4). These constants may be different from line to line. Multiplying (3.1) by d jk (t) and summing the resulting equations over j, we obtain Also, multiplying (3.1) by d jk (t), we get If we take sufficient large k 1 > 0 and use the assumption (A 2 ), we get with some positive λ 1 , relating to the indicated constants in (A 2 ) . We note that The application of Gronwall lemma to (3.10) holds . Also, we differentiate (3.1) with respect to t and get (3.14) Multiplying (3.14) by d jk (t) and summing the resulting equations over j, we obtain in which the Sobolev embedding theorem has been used. Furthermore, the growth condition (2.2) and the Hölder inequality mean that Therefore, we getˆΩ and Then, the applications of the estimates (3.13) and (3.15)- (3.18) give that ∃λ 1 ≥ λ 2 > 0, depending on C 0 , such that Here, assume We show that E 3 (0) is uniformly bounded for k under the conditions in Theorem 3 now. It follows by (3.1) that Especially, suppose t = 0, we get By Young inequality with ε, (3.24) Suppose 0 < ε ≤ 1/6. Then, from (3.22) to (3.24) that Therefore, the inequality (3.20) shows and the estimates (3.13) and (3.26) give that  |g (x, u k (x,t))| α+1 α dxdt is bounded. Accordingly, by Lemma 2 in Chap. 1 [17], we conclude with these convergences, by using the limit in the approximate equation (3.5), we get So, u (t) is a weak solution of (1.1) and supplies (2.5) and (2.6), and the proof of existence for the solution u (t) of (1.1) is completed. We derive the estimates for Pu (t) 2 and Pu t (t) 2 now. Also, we write u instead of u k for convenience and view the estimates for u as a limit of u k . Supposing v = −Pu in (3.31), we obtain with some C 0 > 0 and Also, assuming v = −Pu t in (3.31), we get This means that We note that . We get from (3.32) and (3.39) that Suppose k 1 ≥ 3 and η is small that 1 − η (1 + k 1 /2) ≥ 4/5. Then, (3.40) shows We note that Also (3.41) and (3.43) give that ∃λ 1 β ≥ λ 3 > 0, depending on C 0 , such that Otherwise, we get where the facts k 1 ≥ {4, 2 + c 0 } and Sobolev imbedding theorem (see [17]) have been used. So, by the estimates (3.45) and (3.46) that To establish the uniqueness, we suppose that u (t) and v (t) are two solutions of (1.1), which supply the estimates (2.5)-(2.7) and u (0 Multiplying (3.48) by W, we obtain and where σ 1 = n (α − 1) /2 ≤ 2n/ (n − 2) , σ 2 = n/2. From (2.5) and Sobolev imbedding theorem, there is C 3 > 0 such that Then, Then, the estimates (3.50)-(3.52) indicate that The integral inequality (3.53) represents that there exists T 1 > 0, such that W (t) = 0 in [0, Then, we conduce that . This shows the proof of uniqueness. Now, we establish u ∈ C ([0, ∞) ; H m 0 (Ω)) . Assume t > s ≥ 0. Then, This indicates u (t) ∈ C ([0, ∞) ; H m 0 (Ω)) . Also, we get  This shows that u (t) ∈ C 1 ([0, ∞) ; H m 0 ) and the proof of Theorem 3 is completed.
Proof of Theorem 5 From Lemma 2, it is sufficient to indicate that there exists a continuous operator semigroup {S (t)} on X such that S (t) (u 0 , u 1 ) = (u (t) , u t (t)) for each t ≥ 0. By the estimates (2.7), we accomplish that β 0 = (u, v) ∈ X| P is an absorbing set of {S (t)} t≥0 and for any (u 0 , u 1 ) ∈ X, dist X (S (t) (u 0 , u 1 ) , β 0 ) ≤ C 5 e −λ 3 t , t ≥ 0 (4.22) where the constants C 4 , C 5 are in (2.7). By Theorem 2, S (t) : X → X is continuous and asymptotically compact on X. From a general theory (see [1,11]), we conclude that S (t) admits a global attractor A on X defined by where [D] X is the closure of the set D in X. Then we prove the Theorem 5.

Decay property of solution for (1)
In this section, we search the decay property of solution to (1.1) with f ≡ 0. Firstly, we present a well-known Lemma that will be needed.