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.
Keywords:sixth-order Cahn-Hilliard equation; time-periodic solution; existence; Campanato space
In this paper, we are concerned with the following problem for the sixth-order Cahn-Hilliard type equation:
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
Now, we give some useful lemmas.
Now we consider the following linear periodical problem:
Proof Noticing the condition (2.8) and the boundary value condition (2.9), we use the Poincaré inequality and interpolation method (see Chapter 5 in ) and get
which implies that
Using the Young inequality and (2.12), we obtain
Proof The estimate (2.15) is obvious. In fact, by the Hölder inequality,
Combining the above result with (2.15), it follows that
We discuss it in the following two cases.
Similarly, we can estimate other three terms. Combining the above expressions yields
On the other hand,
Combining the above two yields
Finally, from (2.20), (2.21) and (2.22), we see that
which combined with (2.19) yields
Combining the above with (2.24) yields
Combining the above with (2.23) yields the desired estimate (2.18).
is added. Then following the argument as in Case I, we can complete the proof of (2.18).
Using the interpolation inequality, we have
Replacing R in (2.23) by 2R, and combining the result with the above inequality, we have
On the other hand, by (2.25),
The proof of this lemma can be found in .
By Lemma 2.6, we have
3 The main result and its proof
In this section, we represent the main result of this paper.
To prove the existence of this solution, we employ the Leray-Schauder fixed point theorem which enables us to study the problem by considering the following equation:
subject to the conditions (1.2)-(1.3), where σ is a parameter taking value on the interval , and is periodic in time t with period T, where . For any given function , from linear classical theory (see ), we see that problems (3.1) and (1.2)-(1.3) admit a unique solution , and hence we can define a mapping ℒ as follows:
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.
This result can be directly obtained by a compact embedding theorem, so we omit the details here.
whereCis a constant independent of the solutionuandσ.
then from the Poincaré inequality we know that
which implies that
It follows from (3.5) that
By (1.2), we have
On the other hand, by the Young inequality,
Combining the above expressions, we obtain
Combining the above with (3.6) and (3.7), we see that
On the other hand, integrating by parts and using (1.2), we have
Integrating the above inequalities over and noticing (3.13), we see that the terms of left hand side in these inequalities can all be estimated by and a constant number C. Then by the boundary value condition and (3.10), we have
and also, by the above discussion, we have
By (3.17) and the approach similar to (3.14), we can derive
Now we set
On the other hand, by (3.16), (3.17) and (3.18), we have
Applying the Poincaré inequality and the Friedrichs inequality , we conclude that .
Finally, we set
From the proof of Lemma 3.2, we see that () and can be all uniformly bounded by a constant number C. Therefore the coefficient K in Theorem 2.1 now only depends on the Hölder exponent α. So, for , we have
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 declare that they have no competing interests.
All authors contributed equally to the manuscript and read and approved the final manuscript.
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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