Abstract and Applied Analysis

Volume 2013 (2013), Article ID 717042, 8 pages

http://dx.doi.org/10.1155/2013/717042

Research Article

Nonuniform Continuity of the Osmosis K(2, 2) Equation

Aiyong Chen,1,2,3 Yong Ding,² and Wentao Huang⁴

¹Guangxi Key Laboratory of Trusted Software, Guilin University of Electronic Technology, Guilin 541004, China

²School of Mathematics and Computing Science, Guilin University of Electronic Technology, Guilin 541004, China

³Center of Nonlinear Science Studies, Kunming University of Science and Technology, Kunming, Yunnan 650093, China

⁴Department of Mathematics, Hezhou University, Hezhou, Guangxi 542800, China

Received 5 August 2013; Accepted 4 November 2013

Academic Editor: Henrik Kalisch

Copyright © 2013 Aiyong Chen et al. 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.

Abstract.

The qualitative theory of differential equations is applied to the osmosis K(2, 2) equation. The parametric conditions of existence of the smooth periodic travelling wave solutions are given. We show that the solution map is not uniformly continuous by using the theory of Himonas and Misiolek. The proof relies on a construction of smooth periodic travelling waves with small amplitude.

1. Introduction

It is well known that the study of nonlinear wave equations and their solutions are of great importance in many areas of physics. Travelling wave solution is an important type of solution for the nonlinear partial differential equation and many nonlinear partial differential equations have been found to have a variety of travelling wave solutions.

The well-known Korteweg-de-Vries equation

was first derived by Boussinesq in 1877, and later by Korteweg and de Vries in 1895, as an approximate description of surface water waves propagating in a canal. This equation has since found application to a range of problems in solid and fluid mechanics as well as plasma physics and astrophysics. The KdV equation has smooth solitary wave solutions and smooth periodic wave solutions [1]. Bona and Smith [2] considered the Cauchy problem for (1). Bona [3] investigated the stability of solitary waves of (1). Angulo Pava et al. [4] studied stability of cnoidal waves of (1).

The Camassa-Holm equation

was proposed by Camassa and Holm [5] as a model equation for unidirectional nonlinear dispersive waves in shallow water. This equation has attracted a lot of attention over the past decade due to its interesting mathematical properties. The Camassa-Holm equation has been found to has peakons, cuspons, stumpons, and composite wave solutions [6–11]. Himonas and Misiołek [12] showed that for the solution map for the Camassa-Holm equation is not uniformly continuous from any bounded set in into . A key step in the proof of that result is a construction of a sequence of smooth travelling waves. Himonas et al. [13] extend the result to the range . Their proof is based on the approximation of solutions by terms containing high and low frequencies and exploring the conservation of the norm.

The Degasperis-Procesi equation

was originally derived by Degasperis and Procesi. Zhang and Qiao [14] gave smooth and cusped soliton solutions of the Degasperis-Procesi equation. Liu and Yin [15] proved that the first blowup in finite time to (3) must occur as wave breaking, and shock waves possibly appear afterwards. Christov and Hakkaev [16] considered the problem of the uniformly continuity of Degasperis-Procesi equation.

In 1993, Rosenau and Hyman [17] introduced a genuinely nonlinear dispersive equation, a special type of KdV equation, of the form

where both the convection term and the dispersion effect term are nonlinear. These equations arise in the process of understanding the role of nonlinear dispersion in the formation of structures like liquid drops. If , then there exits special form

When , then (5) becomes the K(2, 2) equation

Rosenau and Hyman derived solutions called compactons for (6). For , Xu and Tian [18] introduced the osmosis K(2, 2) equation

where the negative coefficient of dispersion term denotes the contracting dispersion. They obtained the peaked solitary wave solution and the periodic cusp wave solution for (7). Zhou et al. [19] obtained two new types of travelling wave solutions called kink-like and antikink-like wave solutions. Zhou and Tian [20] obtained the analytic expressions of soliton solution of (7) by using the bifurcation method of dynamical systems. Deng and Han [21] successfully found a peaked wave solution of (7) by using the first-integral method. Deng et al. [22] obtained some new exact travelling-wave solutions and stationary-wave solutions by using the auxiliary elliptic equation method. Recently, Chen and Li [23] obtained single peak solitary wave solutions of the osmosis K(2, 2) equation.

To the best of our knowledge, the problems of the well-posedness and the uniformly continuity of (7) have not yet been considered. Applying Kato’s theory for abstract quasilinear evolution equation of hyperbolic type [16], ones may obtain the local wellposedness for (7). Here, we do not consider the wellposedness for (7). Following [12], we consider the problem of the uniformly continuity of (7) by constructing two sequences of solutions. We hope to extend the result to the range by using approximate solutions and delicate commutator and multiplier estimates in the future. Our main result is the following theorem.

Theorem 1. For any , the solution map for (7) is not uniformly continuous from any bounded set in into , where . More precisely, for each there exist constants and and two sequences of smooth solutions and of (7) such that for any ,

The paper is organized as follows. In Section 2, we discuss the dynamical behavior of solutions of the K(2, 2) equation (5) and give parameter condition of existence of the smooth periodic travelling wave solutions. In Section 3, we provide a precise estimate of periods of the periodic travelling wave solutions. In Section 3, we establish upper bounds for these solutions in -norms. The last section contains the proof of the main result.

2. Dynamical Analysis of Travelling Waves

In this section we investigate the periodic travelling wave solutions of (5). Note that if is a classical solution of (5), then so is the function

If is to be a solution to (5), the function must satisfy the ordinary differential equation

Integrating this equation gives

where is an integration constant. Equation (11) is equivalent to the planar system

with the first integral

where is also an integral constant. As well known, system (12) has a periodic solution if and only if it has a center. Using qualitative theory of differential equations [24, 25], we can easily verify the following statement.

Proposition 2. System (12) has a center if and only if satisfies one of the following parameter set (see Figures 1, 2, 3, 4, 5, and 6):

Figure 1: Phase portrait of system (12) for .

Figure 2: Phase portrait of system (12) for .

Figure 3: Phase portrait of system (12) for .

Figure 4: Phase portrait of system (12) for .

Figure 5: Phase portrait of system (12) for .

Figure 6: Phase portrait of system (12) for .

3. Estimates of Periods of Solutions

In this section we construct a family of smooth travelling wave solutions of suitably high frequency and provide a precise estimate of their periods.

Let and be the minimum and maximum of the function , correspondingly; that is, (see Figure 7). We assume that for sufficiently small . We assume that , since there are similar results for . Equation (13) gives

Expressing through and we find

Then (15) becomes

where

In comparison with Camassa-Holm equation, the function is a quartic function rather than cubic function. It can be seen that (18) admits to a nonconstant solution with period for certain , which satisfies the following initial value problem:

Figure 7: The periodic solution of (18).

The above discussion can be summarized as follows.

Proposition 3. For any , there exists a positive number and even periodic smooth function which solves (18) and (20). The function satisfies

and , where is a travelling wave solution of the osmosis K(2, 2) equation (7).

The next proposition gives precise estimates for the period of the solution in terms of the parameters and .

Proposition 4. The period of function depends continuously on parameters and and satisfies

Proof. The half period can be expressed as

where . The function is increasing in the interval , where . On the other hand,

Thus, and in interval . Then

The estimate for is . Then from (26) it follows the lower bound

Estimating we have . Choose and in such a way to achieve

Then with the help of (25) it follows

Combining (27) and (29), we complete the proof of the proposition.

4. Sobolev Estimates of Solutions

We write for the sake of (22). Since is continuous for sufficiently large, we can find and such that

where . Hence, we have constructed high-frequency solution with the period .

Next we need some estimates in order to obtain upper bounds for these solutions. We start with estimates of the derivatives.

Proposition 5. Suppose . Then for any , there exists a constant such that

For we have

Proof. With the help of (18), the first derivative is estimated as follows:

For using (20) we have

Next, we proceed by induction and assume that (32) is true for all positive integers up to . To estimate -order derivative we have from (20)

Differentiate both sides of (36) and divide by to obtain

Taking derivatives and using Leibniz rule we have

Now, can be expressed as follows:

The last equation together with the induction hypothesis gives

and therefore we have

Thus

which complete the proof.

Further, we proceed with estimates.

Proposition 6. For any , there exists a constant such that

For we have

Proof. For the first derivative we get

For the second derivative using symmetry, periodicity, and integrating by parts, we have

Since , using (17), the last equality gives

Proceeding by induction and using the expression for in Proposition 5, we obtain

Applying the estimates of Proposition 5 and using the induction hypothesis, we estimate further as follows:

Therefore, we have

We close the section with an interpolation argument to obtain the estimates for noninteger values of the Sobolev index in the norm

where is the Fourier transform of .

Proposition 7 (see [12]). Let be the periodic smooth solution constructed in Proposition 3 for any , there exists a constant such that

5. Proof of Main Theorem

Define two sequences of travelling wave solutions

and pick

We show that these sequences are bounded, their difference goes to zero at time zero and stays away from zero at any other time.

It is sufficient to estimate the -norm of since it is bigger than the -norm of . From (53), we have

Also, using (31) and the first of the estimates in (44) of Proposition 6, we find that

Combining this inequality, we get

where . Note that it is already chosen . Hence, both sequences of smooth solutions are bounded. Further,

Finally, the behavior at time can be established as follows:

where denotes the Fourier transform of the function with respect to ; that is,

after a change of variables. Therefore,

Keep only the term that corresponds to gives the inequality

Since is -periodic, we have

Also using the fact that is an even function and integrating by parts, we obtain

Therefore,

where

Then, in the same line as Lemma 4.1 in [12] the integral for can be estimated as

Returning to (65) one gets

Thus, the desired estimate is obtained as in [12] using (31) and .

Acknowledgments

This work is supported by the Foundation of Guangxi Key Lab of Trusted Software, The State Key Laboratory of Integrated Services Networks open grant 2012, Guangxi Key Lab of Wireless Wideband Communication & Signal Processing open grant 2012, Guangxi Natural Science Foundation (no. 2011GXNSFA018134, no. 2013GXNSFBB053005, and no. 2013GXNSFAA019006), and the National Natural Science Foundation of China (no. 11161013, no. 11261013, and no. 11361017). The author wish to thank the anonymous reviewers for their helpful comments and suggestions.

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