Time-periodic solutions for a driven sixth-order Cahn-Hilliard type equation
© Liu et al.; licensee Springer. 2013
Received: 6 November 2012
Accepted: 14 March 2013
Published: 4 April 2013
We study a driven sixth-order Cahn-Hilliard type equation which arises naturally as a continuum model for the formation of quantum dots and their faceting. Based on the Leray-Schauder fixed point theorem, we prove the existence of time-periodic solutions.
MSC:35B10, 35K55, 35K65.
where , , , and are Hölder continuous functions defined on with period T, belongs to the space for some with . Furthermore, we assume that , , , , where γ, ν, , , N, L and Λ are positive constants.
Equation (1.1) with arises naturally as a continuum model for the formation of quantum dots and their faceting; see . It can also be used to describe competition and exclusion of biological population . If we consider that the perturbation function (for example, source) has the influence, then we obtain equation (1.1).
Korzec et al.  studied equation (1.1) with . New types of stationary solutions of one-dimensional driven sixth-order Cahn-Hilliard type equation (1.1) are derived by an extension of the method of matched asymptotic expansions that retains exponentially small terms. Liu et al.  proved that equation (1.1) with possesses a global attractor in the () space, which attracts any bounded subset of in the -norm.
During the past years, many authors have paid much attention to other sixth-order thin film equations such as the existence, uniqueness and regularity of the solutions [5–7]. However, as far as we know, there are few investigations concerned with the time-periodic solutions of equation (1.1), even though there is some literature for population models and Cahn-Hilliard [8, 9]. In fact, it is natural to consider the time-periodic solutions of equation (1.1) when it is used to describe the models of the growth and dispersal in the population which is sensitive to time-periodic factors (for example, seasons). In this paper, we prove the existence of time-periodic solutions of problem (1.1)-(1.3) based on the framework of the Leray-Schauder fixed point theorem which can be found in any standard textbook of PDE (see, for example, ). For this purpose, we first introduce an operator ℒ by considering a linear sixth-order equation with a parameter . After verifying the compactness of the operator and some necessary a priori estimates for the solutions, we then obtain a fixed point of the operator in a suitable functional space with , which is the desired solution of problem (1.1)-(1.3).
The main difficulties for treating problem (1.1)-(1.3) are caused by the nonlinearity of both the fourth-order term and the convective factors. The main method that we use is based on the Schauder-type a priori estimates, which here are obtained by means of a modified Campanato space. We note that the Campanato spaces have been widely used for the discussion of partial regularity of solutions of parabolic systems of second order and fourth order. So, in the following section we give a detailed description and the associated properties of such a space, and subsequently, in the next section we prove the existence of classical time-periodic solutions of problem (1.1)-(1.3).
2 Hölder norm estimates
where denotes the parabolic boundary of and denotes the area of .
Now, we give some useful lemmas.
Lemma 2.1 
Here we simply assume that is sufficiently smooth. Our main purpose is to find the relation between the Hölder norm of the solution u and .
and , are the down-side and up-side points of , and is the boundary of .
Some essential estimates on and are based on the following lemmas.
where C is a positive constant, .
Combining (2.12), (2.13) and (2.14) yields the estimate (2.10) with .
Similarly, multiplying (2.7) by and , we can obtain the estimates (2.10) with , . □
and C is a constant number. Further, (2.15) and (2.16) still hold if is replaced by or .
To prove the results on or , we only need to differentiate equation (2.4) once or twice with respect to x. And the next procedures are completely similar to the above argument. □
We first prove (2.18) in the case . In such a case, , . Choose a smooth function satisfying the following requirements.
If , then the value of for is changed into 1.
If , then the value of for is changed into 1.
Then we prove (2.18) in the case 0 or . Take the case as an example. Choose another smooth function such that when ; when ; ; for all .
is added. Then following the argument as in Case I, we can complete the proof of (2.18).
which together with (2.25) yields (2.17) with .
For (2.17) with and , we should first multiply (2.4) by and respectively, and the remaining parts are similar and easier. □
where C is a constant number. Further, (2.26) still holds, if is replaced by or .
which combined with (2.27) implies (2.26). The proofs of the results on or are similar. □
The proof of this lemma can be found in .
Further, (2.28) still holds if u is replaced by Du or .
Using Lemma 2.1, we immediately obtain (2.28). The proofs of the results on or are similar. □
3 The main result and its proof
In this section, we represent the main result of this paper.
Theorem 3.1 Problem (1.1)-(1.3) admits a time-periodic solution .
Obviously, for any given , . By virtue of the Leray-Schauder fixed point theorem, to prove the existence of solutions of problem (1.1)-(1.3), we only need to show that the mapping ℒ is compact and prove that there exists a constant independent of and σ such that, for any u and σ satisfying , . Moreover, it follows from the above arguments that u is a classical solution. Then we consider the problem in in turn. Finally, we know that initial boundary value problem (1.1)-(1.3) admits a classical solution in Q.
Lemma 3.1 The mapping is compact.
This result can be directly obtained by a compact embedding theorem, so we omit the details here.
where C is a constant independent of the solution u and σ.
By (1.2), we know that .
Applying the Poincaré inequality and the Friedrichs inequality , we conclude that .
By an approach similar to the above argument, we can obtain the last result that . The proof of this lemma is complete. □
which combines with the results of Lemma 3.2. We know that , where C is independent of u and σ. Then, it follows from the results in  that . Recalling the discourse in the beginning of this section, we conclude from the Leray-Schauder fixed point theorem that admits a fixed point u in the space , which is the desired solution of problem (1.1)-(1.3). The proof of Theorem 3.1 is completed. □
The authors would like to express their deep thanks to the referees for their valuable suggestions, for the revision and improvement of the manuscript. This research was partly supported by the Scientific Research Foundation for the Returned Overseas Chinese Scholars, State Education Ministry.
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