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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 414590, 24 pages
Some Properties of Solutions for the Sixth-Order Cahn-Hilliard-Type Equation
Department of Mathematics, Jilin University, Changchun 130012, China
Received 10 July 2012; Revised 18 August 2012; Accepted 6 September 2012
Academic Editor: Ruediger Landes
Copyright © 2012 Zhao Wang and Changchun Liu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
We study the initial boundary value problem for a sixth-order Cahn-Hilliard-type equation which describes the separation properties of oil-water mixtures, when a substance enforcing the mixing of the phases is added. 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 solutions blow up at a finite time. We also discuss the existence of global attractor.
We consider the following equation: where is a bounded domain in with smooth boundary and . Equation (1.1) is supplemented by the boundary value conditions: and the initial value condition:
Equation (1.1) 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 is the scalar order parameter which is proportional to the local difference between oil and water concentrations. The amphiphile concentration is approximated by the quadratic function  From the physical consideration, we prefer to consider a typical case of the volumetric free energy , that is, , in the following form:
During the past years, many authors have paid much attention to the sixth-order-parabolic 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 Zajączkowski  proved that the initial-boundary value problem (1.1)–(1.5) with admits a unique global smooth solution which depends continuously on the initial datum. Schimperna and Pawłow  studied (1.1) with viscous term and logarithmic potential: They investigated the behavior of the solutions to the sixth-order system as the parameter tends to . The uniqueness and regularization properties of the solutions have been discussed. Liu studied the following equation: and he proved the existence of classical solutions for two dimensions . Korzec et al.  established the stationary solutions of the sixth-order convective Cahn-Hilliard type equation, which arose from epitaxial growth of nanostructures on crystal surfaces.
The dynamic properties of (1.1), such as the global asymptotical behaviors of solutions and existence of global attractors, are important for the study of higher order parabolic system. During the past years, many authors have paid much attention to the attractors. Nicolaenko et al.  proved the existence of global compact and finite-dimensional attractors for Cahn-Hilliard equation (see also [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 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 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 space.
The plan of the paper is as follows. In Section 2, we investigate the global existence of the solution when . The blowup of the solution is obtained in Section 3 when . In Section 4, we obtain the existence of the global attractor in space.
Throughout the paper, we use to denote , and The norms of , , and are denoted by , , and .
2. Global Existence
Now, we deal with problem (1.1)–(1.3) for . 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 , , and ,(i)if and , then the problem (1.1)–(1.3) exists a unique global solution ;(ii)if , , and are sufficiently large, then the problem (1.1)–(1.3) also admits a unique global solution .
Proof. (i) First, we set
Integrating by parts and using (1.1) itself and the boundary value condition (1.2), we see that
On the other hand, we have
By Young’s inequality, we derive
Combining the above inequalities, we get
From (2.9), we know
By (2.7), (2.8), and (2.10), we obtain
By Sobolev embedding theorem, it follows from (2.11) that
The second step: multiplying (1.1) by and integrating with respect to , we obtain
Thus, it follows from (2.11) and (2.12) that
By the Gronwall’s inequality, (2.14) implies The third step: multiplying (1.1) by and integrating with respect to , we obtain On the other hand, by the Nirenberg’s inequality and (2.12), we have Using (2.7), (2.12), and the above inequality, we derive From these inequalities we finally arrive at A Gronwall’s argument now gives Similarly, multiplying (1.1) by and using we obtain So that Define the linear spaces and the associated operator , where is determined by the following linear problem: From the discussions above and by the contraction mapping principle, has a unique fixed point , which is the desired solution of the problem (1.1)–(1.3).
Now, we show the uniqueness. Assume and are two solutions of the problem (1.1)–(1.3). Let . Then satisfies where .
Multiplying the above equation by and integrating with respect to , integrating by parts, and using the boundary value condition, we have The Gronwall inequality yields Therefore, ,.
(ii) From the proof of (i), we know which together with the Young inequality gives
Note that that is, Hence,
Taking into account (2.31) and (2.34), we see Hence, when is sufficiently large such that , we obtain the estimates (2.7)–(2.9). The other steps are similar to the proof of (i), so the details are omitted here.
3. Blow Up
In the previous sections, we have seen that the solution of the problem is globally existent, provided that . The following theorem shows that the solution of the problem blows up at a finite time for .
Theorem 3.1. If , is sufficiently large.(i)If , then the solution of (1.1)–(1.3) blows up in finite time, that is, for , (ii)If and , the solution must blow up in a finite time.(iii)If , , and are large enough, the solution blows up in finite time.
Proof. (i) From the proof of Theorem 2.1, we know
Let be the unique solution of the following problem:
It is easily seen that
Multiplying (1.1) by and integrating with respect to , we obtain Owing to , and (3.5), it follows from the above inequality that where . Hence, when is sufficiently large, such that , then by , we know that 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
It follows that
Again by (3.12) and (3.3), we get By , choosing large enough, we obtian Similar to the proof of (i), we easily see that (ii) holds.(iii) The crucial term is By the Young inequality, we have By (3.3) and (3.16), we have It follows that If , substituting the (3.18) into (3.13), similarly, we know that the solution must blow up in finite time.
4. Global Attractor in Space
We let where is a constant, and the on is a well-defined semigroup.
In order to prove Theorem 4.1, we need to establish some a priori estimates for the solution of problem (1.1)–(1.3). In what follows, we always assume that is the semigroup generated by the weak solutions of (1.1) with initial data .
Lemma 4.2. There exists a bounded set whose size depends only on and in such that for all the orbits staring from any bounded set in , , such that for all all the orbits will stay in .
Proof. It suffices to prove that there is a positive constant such that for large , then the following holds: From (2.22), we have On the other hand, we know that Hence, we see that which gives The proof is completed.
Lemma 4.3. For any initial data in any bounded set , there exists such that which turns out that is relatively compact in .
Proof. From Theorem 2.1, we know when or that (2.14) holds. Integrating (2.14) over , we obtain
On the other hand, Differentiating (1.1) gives Multiplying (4.10) by and integrating on , 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 ; the -limit set of absorbing set is a global attractor in . By Lemma 4.3, this global attractor is a bounded set in . 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 th order space . Because both and lead to Theorem 4.1, the following proofs are based on Theorem 4.1, hence for simplicity, we let .
In order to consider the global attractor for (1.1) in space, we introduce the definition as follows: where is a constant.
We rewrite (1.1) as where
The linear operator is a sectorial operator which generates an analytic semigroup . Without loss of generality, we assume that generates the fractional power operators and the fractional order spaces as follows: where is the domain of and is a compact inclusion for any [see Pazy ].
The space is given by the closure of in and for .
Lemma 4.4. Let be the solution of (4.17) and the semigroup generated by (4.17). Assume is the fractional order space generated by and(1)for some there is a bounded set , for any there exists such that (2)there is a , for any bounded set , ∃ and such that Then (4.17) has a global attractor which attracts any bounded set in the -norm.
Lemma 4.5. Let be a sectorial operator which generates an analytic semigroup . If all eigenvalues of satisfy for some real number , then for we have(1) is bounded for all and ;(2), for all ;(3)for each , is bounded, and where some , is a constant only depending on ;(4) the -norm can be defined by
Now, we give the main theorem.
Proof. Owing to (4.17), the solution of the system (1.1)–(1.3) can be written as
First of all, we are going to prove that for any , the solution of the problem is uniformly bounded in , that is, for any bounded set , there exists such that
From Theorem 4.1, we have known that, for any bounded set there is a constant such that Next, according to Lemma 4.4, we prove (4.24) for any in the following steps.
Step 1. We are going to show that for any bounded set , there exists such that In fact, by the embedding theorems of fractional order spaces , we have From (4.23), we have We deduce that which means that is bounded.
Hence, from (4.25), (4.28), (4.29), and Lemma 4.5, we have where .
Step 2. We prove that for any bounded set , there is a constant such that
In fact, by the embedding theorems, we have We deduce that It implies that Therefore, it follows from (4.26) and (4.33) that Thus, by the same method as that in Step 1, we get from (4.35) that where .
Step 3. We prove that for any bounded set , there is a constant such that
In fact, by the embedding theorems, we have We deduce
It implies that Therefore, it follows from (4.31) and (4.40) that Thus, by the same method as those in Steps 1 and 2, we get from (4.41) that where .
Step 4. We prove that for any bounded set , there is a constant such that
In fact, by the embedding theorems, we have Hence, similar to the above, we have It implies that Therefore, it follows from (4.37) and (4.46) that Thus, by the same method as that in Steps 1, 2, and 3, we get from (4.47) that where .
In the same fashion as in the proof of (4.43), by iteration we can prove that for any bounded set , there is a constant such that That is, for all the semigroup generated by the problem (1.1)–(1.3) is uniformly compact in .
Secondly, we are going to show that for any , the problem (1.1)–(1.3) has a bounded absorbing set in ; that is, for any bounded set there are and a constant independent of , such that
For , this follows from Theorem 4.1. Now, we will prove (4.13) for any in the following steps.
Step 1. We will prove that for any , the problem (1.1)–(1.3) has a bounded absorbing set in .
By (4.23), we have Let be the bounded absorbing set of the problem (1.1)–(1.3) in , and is the time such that
On the other hand, it is known that where is the first eigenvalue of For any given and , we have By assertion (3) of Lemma 4.5, it follows from (4.51) and (4.29) that for any , we have where is a constant independent of . Then, we infer from (4.55) and (4.56) that (4.50) holds for all .
Step 2. We will show that for any , the problem (1.1)–(1.3) has a bounded absorbing set in .
By (4.33) and (4.51), we deduce that