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:
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
Let , . For any fixed , we define
Let u be a function defined on , and set
where denotes the parabolic boundary of and denotes the area of .
For any and , define
where . By the space we mean the subset of , each element of which satisfies . For , its norm is defined as
Now, we give some useful lemmas.
Now we consider the following linear periodical problem:
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 .
Let be a fixed point and define
Let u be an arbitrary solution of problem (2.1)-(2.3). We split u on as so that solves the problem
and solves the problem
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.
Lemma 2.2For the solution of problem (2.7)-(2.9), we have
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 ) and get
which implies that
Multiplying equation (2.7) by , integrating the result over and using the boundary value condition (2.9), we have
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 and , we can obtain the estimates (2.10) with , . □
Lemma 2.3For any ,
andCis a constant number. Further, (2.15) and (2.16) still hold if is replaced by or .
Proof The estimate (2.15) is obvious. In fact, by the Hölder inequality,
For (2.16), we only consider the case when , . Integrating equation (2.4) over the region , we have
Integrating the above equation with respect to z over , and then integrating the result with respect to y over , we have
By virtue of the mean value theorem and the Hölder inequality, we see that there exists such that
Combining the above result with (2.15), it follows that
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. □
whereCis a constant and ,
Proof In order to prove (2.17) with , we first prove that
We discuss it in the following two cases.
(I) We first prove (2.18) in the case . In such a case, , . Choose a smooth function satisfying the following requirements.
If , then , when , ,
If , then the value of for is changed into 1.
If , then the value of for is changed into 1.
Multiplying equation (2.4) by and integrating the result over , then using the boundary value condition (2.6), we have
By the Young inequality and the definition of , we have
Similarly, we can estimate other three terms. Combining the above expressions yields
As for , we have
As for , we have
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
We can imitate all the above procedures and derive a similar result on , that is,
where H is a polynomial with respect to χ, , , , and satisfies . Using the Sobolev inequality on , we have
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 . Take the case as an example. Choose another smooth function such that when ; when ; ; for all .
With λ stated in the lemma, we multiply (2.4) by and integrate the result over . Then we can derive equalities similar to the above argument in which is replaced by and a term
is added. Then following the argument as in Case I, we can complete the proof of (2.18).
Now we multiply (2.4) by and follow the above argument. Then we derive the same result on :
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 and , we should first multiply (2.4) by and respectively, and the remaining parts are similar and easier. □
Lemma 2.5For any ,
whereCis a constant number. Further, (2.26) still holds, if is replaced by or .
Proof It suffices to show (2.26) for , otherwise we only need to set . By Lemma 2.3 and Lemma 2.4, we have
Taking , we obtain
On the other hand, by (2.25),
which combined with (2.27) implies (2.26). The proofs of the results on or are similar. □
Lemma 2.6Let be a nonnegative and nondecreasing function satisfying
whereA, B, α, βare positive constants and . Then there exists a constantConly depending onA, B, α, βsuch that
The proof of this lemma can be found in .
Theorem 2.1Let be an appropriately smooth function, and letube the smooth solution of problem (2.1)-(2.3). Then, for any , there exists a coefficientKdepending only onα, , , such that
Further, (2.28) still holds ifuis replaced byDuor .
Proof For any fixed point , consider the function , which is clearly nondecreasing with respect to ρ. By Lemma 2.5,
holds for any . By Lemma 2.2,
By Lemma 2.6, we have
for some . Hence,
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 3.1Problem (1.1)-(1.3) admits a time-periodic solution .
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:
together with its composition with , namely
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 3.1The mapping is compact.
This result can be directly obtained by a compact embedding theorem, so we omit the details here.
Lemma 3.2Let be a time-periodic solution of the equation
subject to the conditions (1.2)-(1.3), where . Then
whereCis a constant independent of the solutionuandσ.
Proof First, let be a time-periodic solution of the problem
then from the Poincaré inequality we know that
Multiplying (3.2) by , integrating the result over and using the condition (1.2), then using the Young inequality and (3.4), we have
which implies that
It follows from (3.5) that
By (1.2), we have
Integrating the above inequality over and using (3.8) together with the Young inequality, 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
where , λ is a positive constant depending only on and N. Then . Integrating over , by (3.9) and (3.10), we get
On the other hand, integrating by parts and using (1.2), we have
Integrating the above equality over and noticing the periodicity of F, we have
Integrating over , using (3.9) and (3.10), we have
By virtue of (3.11) and (3.12), we have . Noticing the definition of , we get
By (1.2), we know that .
In order to prove the rest of this lemma, we need to give a priori estimate on . First, by the Gagliardo-Nirenberg inequality, we can obtain
where denotes the norm on . Regulating the exponents and using the Young inequality for every of the above three expressions, we get
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
Multiplying (3.2) by , integrating the result over , using (3.14), (3.15) and the Young inequality, we get
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
By virtue of (3.19) and (3.20), we have . Noticing the definition of , we get
Applying the Poincaré inequality and the Friedrichs inequality , we conclude that .
Finally, we set
By an approach similar to the above argument, we can obtain the last result that . The proof of this lemma is complete. □
Proof of Theorem 3.1 Now we apply Theorem 2.1 to complete the proof of Theorem 3.1. For the smooth function in Theorem 2.1, let
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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