Abstract and Applied Analysis

Volume 2012 (2012), Article ID 414590, 24 pages

doi:10.1155/2012/414590

Research Article

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

Zhao Wang and Changchun Liu

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.

Abstract.

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.

1. Introduction

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 [1]

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 [9] 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 [10] 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 [11]. Korzec et al. [12] 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. [13] 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

This implies

where

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

Hence,

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

In this section, we will give the existence of the global attractors of the problem (1.1)–(1.3) in any th order space .

First of all, we will prove the existence of attractor for . We define the operator semigroup in space by

where is the solution of the problem (1.1)–(1.3).

We let

where is a constant, and the on is a well-defined semigroup.

Theorem 4.1. For every chosen as above, the semiflow associated with the solution of the problem (1.1)–(1.3) possesses in a global attractor , which attracts all the bounded set in .

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

Let

Then, (4.14) and (4.15) can be rewritten as

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 [18]].

The space is given by the closure of in and for .

The following lemmas which can be found in [19, 20] are crucial to our proof.

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.

Theorem 4.6. For any the problem (1.1)–(1.3) has a global attractor in , and attracts any bounded set of in the norm.

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 [18], 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

where is a constant independent of . Thus, we verify from (4.55) and (4.57) that (4.50) is true for all .

By iteration, we can obtain (4.50) for all . Hence, (1.1)–(1.3) has a bounded absorbing set in for all .

Finally, this theorem follows from (4.24), (4.50), and Lemma 4.4. The proof is completed.

Remark 4.7. The attractors in Theorem 4.6 are the same for all , that is, , for all . Hence, . Theorem 4.6 implies that for any , the solution of the problem (1.1)–(1.3) satisfies that

Acknowledgment

The authors would like to express their deep thanks to the referee’s valuable suggestions for the revision and improvement of the paper.

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