Graduate School of Humanities and Sciences, Nara Women's University, Nara 630-8506, Japan
Received 6 October 2010; Revised 23 December 2010; Accepted 26 January 2011
Academic Editor: M.ย Lakshmanan
Copyright ยฉ 2011 Kyoko Tomoeda. 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.
We consider the initial value problem for the reduced fifth-order KdV-type equation: , , , . This equation is obtained by removing the nonlinear term from the fifth-order KdV equation. We show the existence of the local solution which is real analytic in both time and space variables if the initial data satisfies the condition , for some constant . Moreover, the smoothing effect for this equation is obtained. The proof of our main result is based on the contraction principle and the bootstrap argument used in the third-order KdV equation (K. Kato and Ogawa 2000). The key of the proof is to obtain the estimate of on the Bourgain space, which is accomplished by improving Kenig et al.'s method used in (Kenig et al. 1996).
The KdV hierarchy is well known as the series of the Lax pair formulation [1, 2], which are presented as
These equations describe mathematical models of some water waves [3, 4]. We are interested in the existence theory of the analytic solution and the smoothing effect of the KdV hierarchy. There are some results concerning the analyticity for the third-order KdV equation . To. Kato and Masuda  considered the initial value problem of the following equation:
where is the real analytic and the no growth rate function in . They showed that if the initial data is real analytic, then, the global solution of (1.2) is real analytic in the space variable. Hayashi  also considered (1.2) in which is the polynomial. He showed that if the initial data is analytic and has an analytic continuation to a strip containing the real axis, then, the local solution also has the same property. When , (1.2) becomes the third-order KdV equation . K. Kato and Ogawa  proved that has not only the real analytic solution in both time and space variables but also the smoothing effect.
Recently, it is shown that the nonlinear dispersive equations including the KdV hierarchy have the local analytic solution in the space variable (see ). However, neither the existence of the real analytic solution in both time and space variables nor the smoothing effect is obtained for with , because the bilinear estimate of with cannot be obtained by their method used in .
On the other hand, we may expect that the method used in  can work for the reduced equation given by removing from the higher-order KdV equations with . In this paper, as a starting point for this attempt, we consider the following initial value problem of the reduced fifth-order KdV-type equation:
where we may take all coefficients of the nonlinear terms to be equal to 1 without loss of generality. This equation is obtained by removing the nonlinear term from the original fifth-order KdV equation . Our main purpose is to prove not only the existence of a local real analytic solution of (1.3) in both time and space variables but also the smoothing effect.
Before stating the main result precisely, we introduce the function space introduced by Bourgain (see ): for , define that
and is the Fourier transform of in both and variables; that is,
Our main result is the following theorem.
Theorem 1.1. Let and let . Then, for any such that
there exist a constant and a unique solution of (1.3) satisfying
where is the generator of dilation for the linear part of the equation of (1.3). Moreover this solution becomes real analytic in both time and space variables; that is, there exist the positive constants and such that
holds for all and .
Remark 1.2. The initial data has to be analytic except for but is allowed to have -singularity at . It follows from (1.9) that the singularity of disappears after time passes and the regularity of the local solution of (1.3) reaches real analyticity in both time and space variables; that is, the fifth-order KdV-type equation has the smoothing effect.
Remark 1.3. A typical example of the initial data satisfying the condition (1.7) is given by
The existence results of the higher-order KdV equation are studied by many authors. Saut  and Schwarz  proved that each equation of the KdV hierarchy has a unique global solution in the spatially periodic Sobolev space. Kenig, Ponce, and Vega studied the initial value problem of the higher-order dispersive equation
where and is a polynomial having no constant or linear part. They proved the local well-posedness in the weighted Sobolev space [13, 14]. Recently, Kwon  studied the simplified fifth-order KdV-type equation
which is obtained by removing the term from . He showed the local well-posedness for the IVP of this equation in with . On the other hand, Ta. Kato  proved the following result for .
Well-Posedness Theorem (Ta. Kato) (1)Let
Then, the local well-posedness for the IVP of holds in , where
Then, the global well-posedness for the IVP of holds in .
The plan of this paper is as follows. In Section 2, we give the existence and uniqueness of the local solution of (1.3), which is shown by the contraction argument consisting of Lemmas 2.3โ2.5. In Section 3, we prove Lemma 2.4 which gives the bilinear estimate for in the Bourgain space. Kenig et al.  showed the bilinear estimate for of by estimating the potential which appears in an expression of the Bourgain norm of this term via duality. They divided the domain of integration of the potential into 17 subregions. However, their method of the domain decomposition is consistent with , but not with the fifth-order KdV-type equation. We divide this domain into 30 subregions to derive the bilinear estimate for . In Section 4, we show the analyticity of the solution stated in Theorem 1.1 by the bootstrap argument. The result of this paper is announced in Proceedings of the Japan Academy .
Notation. Let be the Fourier transform in the variable, and let and be the Fourier inverse transform in the and variables, respectively. The Riesz operator Dx and its fractional derivative are defined by
respectively, where . Similarly, is defined by
denotes the commutator relation of two operators given by . denotes the space for with the norm
We use Sobolev spaces with both time and space variables
with the norm . Moreover, denotes the space with the norm . The dual coupling is expressed as . The convolution of and with both space and time variables is denoted by . For the constant appearing in Theorem 1.1, we put
For simplicity we make use of three notations:
2. Existence and Uniqueness of the Solution
In this section, we give the proof of the existence and uniqueness of the solution of (1.3). Let and , and we derive the equation which and satisfy. Since , it follows that
Using (2.1) and the following relations
we have from (1.3)
Using the Leibniz rule and (2.1), we can see that
We will show the existence and uniqueness of the solution of (2.3).
Proposition 2.1. Let
where . Then, for any such that and
there exist a constant and a unique solution of (2.3) satisfying
Remark 2.2. The uniqueness of the solution of (2.3) yields for . Moreover, becomes a solution of (1.3), the uniqueness of which also follows.
To prove this proposition we prepare three lemmas (Lemmas 2.3, 2.4 and 2.5), which play an important role in applying the contraction principle to the following system of the integral equations:
where denotes a cut-off function in satisfying
Lemma 2.3. Let and let
where , , and are constants depending on , , , and .
Proof. Equations (2.13) and (2.14) are obtained by replacing the generator by in the argument given by Kenig et al.  (see ). For the proof of (2.15), we refer to Lemmaโโ2.5 in the study by Ginibre-Tsutsumi-Velo .
Lemma 2.4. Let
where . Then
where is a constant depending on , , and .
We give for the proof of this lemma, in Section 3.
Lemma 2.5. Let
where is a constant depending on , , and .
Proof. This lemma is proved by improving Chen et al.โs argument used in the case where .
Proof of Proposition 2.1. We define
We define a map by and
Let and be positive constants satisfying and
We now show that is a contraction mapping from to itself. According to Lemmas 2.3, 2.4, and 2.5, we have for
for any . Here . By taking a sum over , we have
Since , we have from (2.23)
which implies . Similarly, we have for and
Thus, the mapping is contraction from to itself. We obtain a unique fixed point satisfying
on the time interval and . Uniqueness of the solution is also shown by using Bekiranov et al.โs argument in . This completes the proof.
In this section we prove Lemma 2.4. To prove Lemma 2.4 we prepare the following lemma.
Lemma 3.1 (see ). (1) Let and let . If
where is a constant depending on and . (2) If , then
where is a constant depending on .
Proof of Lemma 2.4. By duality, we have
where is the inner product in . Setting
We split into three regions, :
and then, we split into three regions:
We further split , , and into the following regions:
so that, we have
Now we will estimate , , and . To estimate these terms, we prepare some estimates. By (3.8), (3.9) we obtain
where . Since (3.13) and (3.14) yield
we have by (3.9) and (3.13)โ(3.15)
where . Using (3.11), (3.14), and (3.15), we have
In (), we integrate with respect to and first, then, we use Schwarzโs inequality, Fubiniโs theorem, and note that to have
In (), we integrate with respect to and first, then, we use the same way as in (3.18) to have
In we use the change of variables
We integrate with respect to and first, then, we use the same way as in (3.18) to have
In (), we have by a similar argument to (3.22)
where is the region which is obtained from by the change of variables and . We integrate with respect to and first, then, we use the same way as in (3.18) to have
Now we will get bounds for
By using the following methods, we estimate (3.27). The Case of Since
it follows from (3.2) in Lemma 3.1 with , that
where and . Since , we have
where is some constant. The Case of and We consider . By (3.2), we have
where and . Here we have used (3.16) and (3.17) with the change of variables . Noting
Thus, (3.27) is bounded by
in . In the same manner as (3.31)-(3.33), (3.27) are bounded by in , , and .The Case of , We consider . Since
Noting that , we have