Abstract
We study a driven sixthorder CahnHilliard type equation which arises naturally as a continuum model for the formation of quantum dots and their faceting. Based on the LeraySchauder fixed point theorem, we prove the existence of timeperiodic solutions.
MSC: 35B10, 35K55, 35K65.
Keywords:
sixthorder CahnHilliard equation; timeperiodic solution; existence; Campanato space1 Introduction
In this paper, we are concerned with the following problem for the sixthorder CahnHilliard type equation:
where
Equation (1.1) with
Korzec et al.[3] studied equation (1.1) with
During the past years, many authors have paid much attention to other sixthorder
thin film equations such as the existence, uniqueness and regularity of the solutions
[57]. However, as far as we know, there are few investigations concerned with the timeperiodic
solutions of equation (1.1), even though there is some literature for population models
and CahnHilliard [8,9]. In fact, it is natural to consider the timeperiodic 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 timeperiodic factors (for example, seasons). In this paper,
we prove the existence of timeperiodic solutions of problem (1.1)(1.3) based on
the framework of the LeraySchauder fixed point theorem which can be found in any
standard textbook of PDE (see, for example, [10]). For this purpose, we first introduce an operator ℒ by considering a linear sixthorder
equation with a parameter
The main difficulties for treating problem (1.1)(1.3) are caused by the nonlinearity of both the fourthorder term and the convective factors. The main method that we use is based on the Schaudertype 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 timeperiodic solutions of problem (1.1)(1.3).
2 Hölder norm estimates
Let
Let u be a function defined on
where
For any
where
Now, we give some useful lemmas.
Lemma 2.1[11]
Let
where
Now we consider the following linear periodical problem:
Here we simply assume that
Let
Let u be an arbitrary solution of problem (2.1)(2.3). We split u on
and
where
and
Some essential estimates on
Lemma 2.2For the solution
whereCis a positive constant,
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 [12]) and get
which implies that
Multiplying equation (2.7) by
Using the Young inequality and (2.12), we obtain
Combining (2.12), (2.13) and (2.14) yields the estimate (2.10) with
Similarly, multiplying (2.7) by
Lemma 2.3For any
where
andCis a constant number. Further, (2.15) and (2.16) still hold if
Proof The estimate (2.15) is obvious. In fact, by the Hölder inequality,
For (2.16), we only consider the case when
Integrating the above equation with respect to z over
By virtue of the mean value theorem and the Hölder inequality, we see that there exists
Combining the above result with (2.15), it follows that
To prove the results on
Lemma 2.4
whereCis a constant and
Proof In order to prove (2.17) with
We discuss it in the following two cases.
(I) We first prove (2.18) in the case
If
If
If
Multiplying equation (2.4) by
By the Young inequality and the definition of
Similarly, we can estimate other three terms. Combining the above expressions yields
As for
As for
that is,
On the other hand,
Combining the above two yields
Notice that
that is,
Finally, from (2.20), (2.21) and (2.22), we see that
which combined with (2.19) yields
We can imitate all the above procedures and derive a similar result on
where H is a polynomial with respect to χ,
Combining the above with (2.24) yields
Combining the above with (2.23) yields the desired estimate (2.18).
(II) Then we prove (2.18) in the case 0 or
With λ stated in the lemma, we multiply (2.4) by
is added. Then following the argument as in Case I, we can complete the proof of (2.18).
Now we multiply (2.4) by
Using the interpolation inequality, we have
Replacing R in (2.23) by 2R, and combining the result with the above inequality, we have
which together with (2.25) yields (2.17) with
For (2.17) with
Lemma 2.5For any
whereCis a constant number. Further, (2.26) still holds, if
Proof It suffices to show (2.26) for
Taking
On the other hand, by (2.25),
which combined with (2.27) implies (2.26). The proofs of the results on
Lemma 2.6Let
whereA, B, α, βare positive constants and
The proof of this lemma can be found in [13].
Theorem 2.1Let
Further, (2.28) still holds ifuis replaced byDuor
Proof For any fixed point
holds for any
Thus,
By Lemma 2.6, we have
for some
Using Lemma 2.1, we immediately obtain (2.28). The proofs of the results on
3 The main result and its proof
In this section, we represent the main result of this paper.
Theorem 3.1Problem (1.1)(1.3) admits a timeperiodic solution
To prove the existence of this solution, we employ the LeraySchauder 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
together with its composition with
Obviously, for any given
Lemma 3.1The mapping
This result can be directly obtained by a compact embedding theorem, so we omit the details here.
Lemma 3.2Let
subject to the conditions (1.2)(1.3), where
whereCis a constant independent of the solutionuandσ.
Proof First, let
then from the Poincaré inequality we know that
Multiplying (3.2) by
which implies that
Moreover,
It follows from (3.5) that
By (1.2), we have
Integrating the above inequality over
that is,
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
Set
where
On the other hand, integrating by parts and using (1.2), we have
Integrating the above equality over
Integrating
By virtue of (3.11) and (3.12), we have
By (1.2), we know that
In order to prove the rest of this lemma, we need to give a priori estimate on
where
Integrating the above inequalities over
and also, by the above discussion, we have
Multiplying (3.2) by
that is,
By (3.17) and the approach similar to (3.14), we can derive
Now we set
Obviously,
On the other hand, by (3.16), (3.17) and (3.18), we have
By virtue of (3.19) and (3.20), we have
Applying the Poincaré inequality and the Friedrichs inequality [15], we conclude that
Finally, we set
By an approach similar to the above argument, we can obtain the last result that
Proof of Theorem 3.1 Now we apply Theorem 2.1 to complete the proof of Theorem 3.1. For the smooth function
From the proof of Lemma 3.2, we see that
which combines with the results of Lemma 3.2. We know that
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the manuscript and read and approved the final manuscript.
Acknowledgements
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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