Some Properties of Solutions for the Sixth-Order Cahn-Hilliard-Type Equation

and Applied Analysis 3 solutions blow up at a finite time. We also discuss the existence of global attractor. We will use the regularity estimates for the linear semigroups, combining with the iteration technique and the classical existence theorem of global attractors, to prove that the problem 1.1 – 1.3 possesses a global attractor inH k ≥ 0 space. The plan of the paper is as follows. In Section 2, we investigate the global existence of the solution when γ1 > 0. The blowup of the solution is obtained in Section 3 when γ1 < 0. In Section 4, we obtain the existence of the global attractor in H k ≥ 0 space. Throughout the paper, we use QT to denote Ω × 0, T , and H6,1 QT { u; ∂u ∂t ∈ L2 QT , Du ∈ L2 QT , 0 ≤ i ≤ 6 } . 1.8 The norms of L∞ Ω , L2 Ω , and H Ω are denoted by ‖ · ‖∞, ‖ · ‖, and ‖ · ‖s. 2. Global Existence Now, we deal with problem 1.1 – 1.3 for n 2, 3. The one-dimensional case is similar. From the classical approach, it is not difficult to conclude that the problem admits a unique classical solution local in time. So it is sufficient to make a priori estimates. Theorem 2.1. For the initial data u0 ∈ H3 Ω , ∂u0/∂n |∂Ω 0, and T > 0, i if γ1 > 0 and a2 > 0, then the problem 1.1 – 1.3 exists a unique global solution u ∈ H6,1 QT ; ii if γ1 > 0, a2 < 0, and γ are sufficiently large, then the problem 1.1 – 1.3 also admits a unique global solution u ∈ H6,1 QT . Proof. i First, we set


Introduction
We consider the following equation: where Ω is a bounded domain in R n n ≤ 3 with smooth boundary and γ > 0. Equation  .1 describes dynamics of phase transitions in ternary oil-water-surfactant systems 1-3 . The surfactant has a character that one part of it is hydrophilic and the other lipophilic is called amphiphile. In the system, almost pure oil, almost pure water, and microemulsion which consist of a homogeneous, isotropic mixture of oil and water can coexist in equilibrium. Here u x, t is the scalar order parameter which is proportional to the local difference between oil and water concentrations. The amphiphile concentration a u is approximated by the quadratic function 1 a u a 2 u 2 a 0 . 1.4 From the physical consideration, we prefer to consider a typical case of the volumetric free energy F u , that is, F u f u , in the following form: During the past years, many authors have paid much attention to the sixth-orderparabolic equation, such as the existence, uniqueness, and regularity of the solutions 4-8 . However, as far as we know, there are few investigations concerned with the sixth order Cahn-Hilliard equation. Pawłow and Zajaczkowski 9 proved that the initial-boundary value problem 1.1 -1.5 with γ 1 1 admits a unique global smooth solution which depends continuously on the initial datum. Schimperna and Pawłow 10 studied 1.1 with viscous term Δu t and logarithmic potential: They investigated the behavior of the solutions to the sixth-order system as the parameter γ tends to 0. The uniqueness and regularization properties of the solutions have been discussed. Liu studied the following equation:  14-16 . In this paper, we consider the problem 1.1 -1.3 . The purpose of the present paper is devoted to the investigation of properties of solutions with γ 1 not restricted to be positive. We first discuss the regularity. We show that the solutions might not be classical globally. In other words, in some cases, the classical solutions exist globally, while in some other cases, such Abstract and Applied Analysis 3 solutions blow up at a finite time. We also discuss the existence of global attractor. We will use the regularity estimates for the linear semigroups, combining with the iteration technique and the classical existence theorem of global attractors, to prove that the problem 1.1 -1.3 possesses a global attractor in H k k ≥ 0 space.
The plan of the paper is as follows. In Section 2, we investigate the global existence of the solution when γ 1 > 0. The blowup of the solution is obtained in Section 3 when γ 1 < 0. In Section 4, we obtain the existence of the global attractor in H k k ≥ 0 space.
Throughout the paper, we use Q T to denote Ω × 0, T , and The norms of L ∞ Ω , L 2 Ω , and H s Ω are denoted by · ∞ , · , and · s .

Global Existence
Now, we deal with problem 1.1 -1.3 for n 2, 3. The one-dimensional case is similar. From the classical approach, it is not difficult to conclude that the problem admits a unique classical solution local in time. So it is sufficient to make a priori estimates. ii if γ 1 > 0, a 2 < 0, and γ are sufficiently large, then the problem 1.1 -1.3 also admits a unique global solution u ∈ H 6,1 Q T .
Integrating by parts and using 1.1 itself and the boundary value condition 1.2 , we see that

4 Abstract and Applied Analysis
This implies On the other hand, we have By Young's inequality, we derive Combining the above inequalities, we get From 2.9 , we know sup 0<t<T Ω u 2 dx ≤ C.

2.12
Abstract and Applied Analysis 5 The second step: multiplying 1.1 by Δu and integrating with respect to x, we obtain

2.13
Thus, it follows from 2.11 and 2.12 that

2.14
By the Gronwall's inequality, 2.14 implies The third step: multiplying 1.1 by Δ 2 u and integrating with respect to x, we obtain

6 Abstract and Applied Analysis
On the other hand, by the Nirenberg's inequality and 2.12 , we have

2.17
Using 2.7 , 2.12 , and the above inequality, we derive

2.18
From these inequalities we finally arrive at

2.19
A Gronwall's argument now gives

2.20
Abstract and Applied Analysis 7 Similarly, multiplying 1.1 by Δ 3 u and using Define the linear spaces and the associated operator W : X → X, u → w, where w is determined by the following linear problem:

2.26
From the discussions above and by the contraction mapping principle, W has a unique fixed point u, which is the desired solution of the problem 1.1 -1.3 .

Abstract and Applied Analysis
Now, we show the uniqueness. Assume u and v are two solutions of the problem where ϕ u −a u Δu − a u /2 |∇u| 2 f u . Multiplying the above equation by w and integrating with respect to x, integrating by parts, and using the boundary value condition, we have
ii From the proof of i , we know which together with the Young inequality gives

2.33
Abstract and Applied Analysis 9 Hence, Taking into account 2.31 and 2.34 , we see

2.35
Hence, when γ is sufficiently large such that γ/2 − C 1 > 0, we obtain the estimates 2.7 -2.9 . The other steps are similar to the proof of i , so the details are omitted here.

Blow Up
In the previous sections, we have seen that the solution of the problem is globally existent, provided that γ 1 > 0. The following theorem shows that the solution of the problem blows up at a finite time for γ 1 < 0.
i If a 2 < 0, then the solution u of 1.1 -1.3 blows up in finite time, that is, for T > 0, ii If a 2 > 0 and a 0 > 0, the solution must blow up in a finite time.
iii If a 2 > 0, a 0 < 0, and γ are large enough, the solution blows up in finite time.
Proof. i From the proof of Theorem 2.1, we know Hence, Let w be the unique solution of the following problem: It is easily seen that Multiplying 1.1 by w and integrating with respect to x, we obtain Owing to a 2 < 0, γ 1 < 0, and 3.5 , it follows from the above inequality that where C 1 > 0. Hence, when −E 0 is sufficiently large, such that −4E 0 − C h 0 , γ 1 , |Ω| ≥ 0, then by γ 1 < 0, we know that u has to blow up.
From the proof of i , we obtain On the other hand, we have From the above inequality, we know Using 3.3 and 3.10 , we see that

3.11
It follows that Again by 3.12 and 3.3 , we get
iii The crucial term is Ω a 0 |∇u| 2 dx.

3.15
By the Young inequality, we have By 3.3 and 3.16 , we have

3.17
It follows that If 2γ a 0 > 0, substituting the 3.18 into 3.13 , similarly, we know that the solution must blow up in finite time.

Global Attractor in H k Space
In this section, we will give the existence of the global attractors of the problem 1.1 -1.3 in any kth order space H k Ω . First of all, we will prove the existence of attractor for γ 1 > 0. We define the operator where u t is the solution of the problem 1.1 -1.3 .

Abstract and Applied Analysis 13
We let where m > 0 is a constant, and the {S t } on X m is a well-defined semigroup. In order to prove Theorem 4.1, we need to establish some a priori estimates for the solution u of problem 1.1 -1. 3 . In what follows, we always assume that {S t } t≥0 is the semigroup generated by the weak solutions of 1.1 with initial data u 0 ∈ H 3 Ω .

Lemma 4.2.
There exists a bounded set B m whose size depends only on m and Ω in X m such that for all the orbits staring from any bounded set B in X m , ∃t 1 t 1 X ≥ 0, such that for all t ≥ t 1 all the orbits will stay in B m .
Proof. It suffices to prove that there is a positive constant C such that for large t, then the following holds: On the other hand, we know that

4.5
Hence, we see that The proof is completed.
14 Abstract and Applied Analysis Lemma 4.3. For any initial data u 0 in any bounded set B ⊂ X m , there exists t 2 t 2 B > 0 such that which turns out that t≥t 2 u t is relatively compact in X m .
Proof. From Theorem 2.1, we know when γ 1 > 0, a 2 > 0 or γ 1 > 0, a 2 < 0 that 2.14 holds. Integrating 2.14 over t, t 1 , we obtain On the other hand, Differentiating 1.1 gives Multiplying 4.10 by Δ 3 u and integrating on Ω, using the boundary conditions, we obtain From 4.11 , 4.9 , and the uniform Gronwall inequality, we have Then by 17, Theorem I.1.1 , we immediately conclude that A m ω B m ; the ω-limit set of absorbing set B m is a global attractor in X m . By Lemma 4.3, this global attractor is a bounded set in H 3 Ω . Thus, we complete the proof of Theorem 4.1.
Secondly, we consider the existence of the global attractors of the problem 1.1 -1.3 in any kth order space H k Ω . Because both γ 1 > 0, a 2 > 0 and γ 1 > 0, a 2 < 0 lead to Theorem 4.1, the following proofs are based on Theorem 4.1, hence for simplicity, we let γ 1.
In order to consider the global attractor for 1.1 in H k space, we introduce the definition as follows: The linear operator L is a sectorial operator which generates an analytic semigroup e tL . Without loss of generality, we assume that L generates the fractional power operators L α and the fractional order spaces H α as follows: where H α D L α is the domain of L α and H β ⊂ H α is a compact inclusion for any β > α see Pazy 18 .
The space H 1/6 is given by H 1/6 the closure of H 1/2 in H 1 Ω and H k H 6k ∩ H 1 for k ≥ 1.
The following lemmas which can be found in 19, 20 are crucial to our proof.
where some σ > 0, C α > 0 is a constant only depending on α; 4 the H α -norm can be defined by Now, we give the main theorem. First of all, we are going to prove that for any α ≥ 0, the solution u t, u 0 of the problem is uniformly bounded in H α , that is, for any bounded set U ⊂ H α , there exists C such that From Theorem 4.1, we have known that, for any bounded set U ⊂ H 1/2 there is a constant C > 0 such that Next, according to Lemma 4.4, we prove 4.24 for any α > 1/2 in the following steps.
Step 1. We are going to show that for any bounded set U ⊂ H α 0 < α < 1 , there exists C > 0 such that In fact, by the embedding theorems of fractional order spaces 18 , we have
Step 2. We prove that for any bounded set U ⊂ H α 2/3 ≤ α < 5/6 , there is a constant C > 0 such that In fact, by the embedding theorems, we have

22
Abstract and Applied Analysis In the same fashion as in the proof of 4.43 , by iteration we can prove that for any bounded set U ⊂ H α α > 0 , there is a constant C > 0 such that That is, for all α ≥ 0 the semigroup S t generated by the problem 1.1 -1.3 is uniformly compact in H α . Secondly, we are going to show that for any α ≥ 0, the problem 1.1 -1.3 has a bounded absorbing set in H α ; that is, for any bounded set U ⊂ H α α ≥ 0 there are T > 0 and a constant C > 0 independent of u 0 , such that 4.50 For α 1/2, this follows from Theorem 4.1. Now, we will prove 4.13 for any α ≥ 1/2 in the following steps.
Step 2. We will show that for any 2/3 ≤ α < 5/6, the problem 1.1 -1.3 has a bounded absorbing set in H α . where C > 0 is a constant independent of u 0 . Thus, we verify from 4.55 and 4.57 that 4.50 is true for all 2/3 ≤ α < 5/6. By iteration, we can obtain 4.50 for all α ≥ 0. Hence, 1.1 -1.3 has a bounded absorbing set in H α for all α ≥ 0.
Finally, this theorem follows from 4.24 , 4.50 , and Lemma 4.4. The proof is completed.

24
Abstract and Applied Analysis Remark 4.7. The attractors A α ⊂ H α in Theorem 4.6 are the same for all α ≥ 0, that is, A α A, for all α ≥ 0. Hence, A ⊂ C ∞ Ω . Theorem 4.6 implies that for any u 0 ∈ H, the solution u t, u 0 of the problem 1.1 -1.3 satisfies that lim t → ∞ inf v∈A u t, u 0 − v C k 0, ∀k ≥ 1. 4.58