Discrete Dynamics in Nature and Society

Volume 2013 (2013), Article ID 163070, 8 pages

http://dx.doi.org/10.1155/2013/163070

Research Article

Persistence Property and Asymptotic Description for DGH Equation with Strong Dissipation

Ke-chuang Wang

Department of Basic, Zhejiang Dongfang Vocational Technical College, Wenzhou, Zhejiang 325011, China

Received 9 February 2013; Accepted 10 March 2013

Academic Editor: Yonghui Xia

Copyright © 2013 Ke-chuang Wang. 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 present work is mainly concerned with the Dullin-Gottwald-Holm (DGH) equation with strong dissipation. We establish a sufficient condition to guarantee global-in-time solutions, then present persistence property for the Cauchy problem, and describe the asymptotic behavior of solutions for compactly supported initial data.

1. Introduction

Dullin et al. [1] derived a new equation describing the unidirectional propagation of surface waves in a shallow water regime:

where the constants and are squares of length scales and the constant is the critical shallow water wave speed for undisturbed water at rest at spatial infinity. Since this equation is derived by Dullin, Gottwald, and Holm, in what follows, we call this new integrable shallow water equation (1) DGH equation.

If , (1) becomes the well-known KdV equation, whose solutions are global as long as the initial data is square integrable. This is proved by Bourgain [2]. If and , (1) reduces to the Camassa-Holm equation which was derived physically by Camassa and Holm in [3] by approximating directly the Hamiltonian for Euler's equations in the shallow water regime, where represents the free surface above a flat bottom. The properties about the well-posedness, blow-up, global existence, and propagation speed have already been studied in recent works [4–13], and the generalized version of a family of dispersive equations related to Camassa-Holm equation was discussed in [14].

It is very interesting that (1) preserves the bi-Hamiltonian structure and has the following two conserved quantities:

Recently, in [15], local well-posedness of strong solutions to (1) was established by applying Kato's theory [16], and some sufficient conditions were found to guarantee finite time blow-up phenomenon. Moreover, Zhou [17] found the best constants for two convolution problems on the unit circle via variational method and applied the best constants on (1) to give some blow-up criteria. Later, Zhou and Guo improved the results and got some new criteria for wave breaking [18].

In general, it is quite difficult to avoid energy dissipation mechanism in the real world. Ghidaglia [19] studied the long time behavior of solutions to the weakly dissipative KdV equation as a finite dimensional dynamic system. Moreover, some results on blow-up criteria and the global existence condition for the weakly dissipative Camassa-Holm equation are presented in [20], and very related work can be found in [21, 22]. In this work, we are interested in the following model, which can be viewed as the DGH equation with dissipation

where , ,   is the weakly dissipative term and is a positive dissipation parameter. Set , then the operator can be expressed by

for all with . With this in hand, we can rewrite (3) as a quasilinear equation of hyperbolic type

It is the dissipative term that causes the previous conserved quantities and to be no longer conserved for (3), and this model could also be regarded as a model of a type of a certain rate-dependent continuum material called a compressible second grade fluid [23]. Our consideration is based on this fact. Furthermore, we will show how the dissipation term affects the behavior of solutions in our forthcoming paper. As a whole, the current dissipation model is of great importance mathematically and physically, and it is worthy of being considered. In what follows, we assume that and just for simplicity. Since is bounded by its -norm, a general case with does not change our results essentially, but it would lead to unnecessary technical complications. So the above equation is reduced to a simpler form as follows:

where

The rest of this paper is organized as follows. In Section 2, we list the local well-posedness theorem for (6) with initial datum ,   and collect some auxiliary results. In Section 3, we establish the condition for global existence in view of the initial potential. Persistence properties of the strong solutions are explored in Section 4. Finally, in Section 5, we give a detailed description of the corresponding solution with compactly supported initial data.

2. Preliminaries

In this section, we make some preparations for our consideration. Firstly, the local well-posedness of the Cauchy problem of (6) with initial data with can be obtained by applying Kato's theorem [16]. More precisely, we have the following local well-posedness result.

Theorem 1. Given that , , there exist and a unique solution to (6), such that

Moreover, the solution depends continuously on the initial data; that is, the mapping is continuous, and the maximal time of existence is independent of .

Proof. Set ,  ,  ,,   , and . Applying Kato's theory for abstract quasilinear evolution equation of hyperbolic type, we can obtain the local well-posedness of (6) in , and .

The maximal value of in Theorem 1 is called the lifespan of the solution in general. If , that is , we say that the solution blows up in finite time, otherwise, the solution exists globally in time. Next, we show that the solution blows up if and only if its first-order derivative blows up.

Lemma 2. Given that ,   , the solution of (3) blows up in finite time if and only if

Proof. We first assume that for some ,   . Equation (6) can be written into the following form in terms of

Multiplying (10) by , and integrating by parts, we have

Differentiating (10) with respect to the spatial variable , then multiplying by , and integrating by parts again, we obtain

Summarizing (11) and (12), we obtain

If is bounded from below on , for example, , is a positive constant, then we get by (13) and Gronwall's inequality the following:

where . Therefore, the -norm of the solution to (10) does not blow up in finite time. Furthermore, similar argument shows that the -norm with does not blow up either in finite time. Consequently, this theorem can be proved by Theorem 1 and simple density argument for all .

Lemma 3. Let , then as long as the solution given by Theorem 1 exists, for any , one has

where the norm is defined as

Proof. Multiplying both sides of (10) by and integrating by parts on , we get

Note that

Then, we have

Hence,

Thus, we easily get

and, therefore,

By integration from to , we get

Hence, the lemma is proved.

We also need to introduce the standard particle trajectory method for later use. Consider now the following initial value problem as follows:

where is the solution to (6) with initial data , and is the maximal time of existence. By direct computation, we have

Then,

which means that is a diffeomorphism of the line for every . Consequently, the -norm of any function is preserved under the family of the diffeomorphism , that is,

Similarly,

Moreover, one can verify the following important identity for the strong solution in its lifespan:

We get that

where is defined by , for in its lifespan.

From the expression of in terms of , for all , , we can rewrite and as follows:

from which we get that

3. Global Existence

It is shown that it is the sign of initial potential not the size of it that can guarantee the global existence of strong solutions.

Theorem 4. Assume that ,  , and   satisfies

for some point . Then, the solution to (6) exists globally in time.

Proof. From the hypothesis and (30), we obtain that , ; , . According to (31) and (32), one can get that when ,

it follows that

that is, is bounded below. Similarly, when ,

so . We also get the bounded below result as above. Therefore, the theorem is proved by Lemma 2.

Corollary 5. Assume that ,  , and   is of one sign, then the corresponding solution to (6) exists globally.

In fact, if is regarded as , we prove this corollary immediately from Theorem 4.

4. Persistence Properties

In this section, we will investigate the following property for the strong solutions to (6) in -space which behave algebraically at infinity as their initial profiles do. The main idea comes from the recent work of Himonas and his collaborators [7].

Theorem 6. Assume that for some and ,    is a strong solution of the initial value problem associated to (6), and that satisfies

for some and . Then,

uniformly in the time interval .

Proof. The proof is organized as follows. Firstly, we will estimate and . Then, we apply the weight function to obtain the desired result. In the following proof, we denote some constants by ; they may be different from instance to instance, changing even within the same line.

Multiplying (6) by with , then integrating both sides with respect to variable, we can get

The first term of the above identity is

and the estimates of the second term is

In view of Hölder's inequality, we can obtain the following estimate for the third term in (39)

For the last term

putting all the inequalities above into (39) yields

Using the Sobolev embedding theorem, there exists a constant

such that we have by applying Gronwall's inequality

For any , we know that

Taking the limits in (46) (notice that and ) from (47), we get

Then, differentiating (6) with respect to variable produces the following equation:

Again, multiplying (49) by with , integrating the result in variable, and considering the second term and the third term in the above identity with integration by parts, one gets

so, we have

Similarly, the following inequality holds

and therefore as before, we obtain

Taking the limits in(53), we obtain

Next, we will introduce the weight function to get our desired result. This function with is independent of as the following:

From (6) and (49), we get the following two equations:

We need some tricks to deal with the following term as in [18]:

where we have used the fact , a.e. . Similar technique is used for the term . Hence, as in the weightless case, we get the following inequality in view of (48) and (54) as follows:

On the other hand, a simple calculation shows that there exists , depending only on and such that for any ,

Therefore, for any appropriate function , one obtains that

and similarly, . Using the same method, we can estimate the following two terms

Therefore, it follows that there exists a constant such that

Hence, the following inequality is obtained for any and any :

Finally, taking the limit as goes to infinity in the above inequality, we can find that for any ,

which completes the proof of Theorem 6.

5. Asymptotic Description

The following result is to give a detailed description on the corresponding strong solution in its lifespan with being compactly supported.

Theorem 7. Assume that the initial datum with is compactly supported in , then the corresponding solution to (6) has the following property: for any ,

where is defined by (24) and is its lifespan. Furthermore, and denote continuous nonvanishing functions, with and for . Moreover, is a strictly increasing function, while is strictly decreasing.

Remark 8. This is an interesting phenomenon for our model; it implies that the strong solution does not have compact -support for any in its lifespan anymore, although the corresponding is compactly supported. No matter that the initial profile is (no matter it is positive or negative), for any in its lifespan, the nontrivial solution is always positive at infinity and negative at negative infinity. Moreover, we found that the dissipative coefficient does not affect this behavior.

Proof. First, since has a compact support, so does . Equation (30) tells us that is compactly supported in in its lifespan. Hence, the following functions are well defined

with

And by integration by parts.

Then, for , we have

where (66) is used.

Similarly, when , we get

Because has a compact support in in the interval for any , we get , for or . Hence, as consequences of (68) and (69), we have

On the other hand,

Substituting the identity (10) into , we obtain

where we used (70). Therefore, in the lifespan of the solution, we have that is an increasing function with ; thus, it follows that for ; that is,

By similar argument, one can verify that the following identity for is true:

In order to finish the proof, it is sufficient to let , and to let , respectively.

References

H. R. Dullin, G. A. Gottwald, and D. D. Holm, “An integrable shallow water equation with linear and nonlinear dispersion,” Physical Review Letters, vol. 87, no. 19, Article ID 194501, 4 pages, 2001.

J. Bourgain, “Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations, II. The KdV equation,” Geometric and Functional Analysis, vol. 3, no. 3, pp. 209–262, 1993.

R. Camassa and D. D. Holm, “An integrable shallow water equation with peaked solitons,” Physical Review Letters, vol. 71, no. 11, pp. 1661–1664, 1993.

A. Constantin, “Finite propagation speed for the Camassa-Holm equation,” Journal of Mathematical Physics, vol. 46, no. 2, Article ID 023506, 4 pages, 2005.

A. Constantin and J. Escher, “Wave breaking for nonlinear nonlocal shallow water equations,” Acta Mathematica, vol. 181, no. 2, pp. 229–243, 1998.

A. Constantin and J. Escher, “Well-posedness, global existence, and blowup phenomena for a periodic quasi-linear hyperbolic equation,” Communications on Pure and Applied Mathematics, vol. 51, no. 5, pp. 475–504, 1998.

A. A. Himonas, G. Misiołek, G. Ponce, and Y. Zhou, “Persistence properties and unique continuation of solutions of the Camassa-Holm equation,” Communications in Mathematical Physics, vol. 271, no. 2, pp. 511–522, 2007.

H. P. McKean, “Breakdown of the Camassa-Holm equation,” Communications on Pure and Applied Mathematics, vol. 57, no. 3, pp. 416–418, 2004.

Z. Xin and P. Zhang, “On the weak solutions to a shallow water equation,” Communications on Pure and Applied Mathematics, vol. 53, no. 11, pp. 1411–1433, 2000.

Y. Zhou, “Wave breaking for a shallow water equation,” Nonlinear Analysis: Theory, Methods & Application, vol. 57, no. 1, pp. 137–152, 2004.

Y. Zhou, “Wave breaking for a periodic shallow water equation,” Journal of Mathematical Analysis and Applications, vol. 290, no. 2, pp. 591–604, 2004.

Y. Zhou, “Stability of solitary waves for a rod equation,” Chaos, Solitons and Fractals, vol. 21, no. 4, pp. 977–981, 2004.

Y. Zhou, “Local well-posedness and blow-up criteria of solutions for a rod equation,” Mathematische Nachrichten, vol. 278, no. 14, pp. 1726–1739, 2005.

L. Ni and Y. Zhou, “Wave breaking and propagation speed for a class of nonlocal dispersive -equations,” Nonlinear Analysis: Real World Applications, vol. 12, no. 1, pp. 592–600, 2011.

L. Tian, G. Gui, and Y. Liu, “On the well-posedness problem and the scattering problem for the Dullin-Gottwald-Holm equation,” Communications in Mathematical Physics, vol. 257, no. 3, pp. 667–701, 2005.

T. Kato, “Quasi-linear equations of evolution, with applications to partial differential equations,” in Spectral Theory and Differential Equations: Proceedings of the Symposium Held at Dundee, Scotland, 1–19 July 1974, vol. 448 of Lecture Notes in Mathetics, pp. 25–70, Springer, Berlin, Germany, 1975.

Y. Zhou, “Blow-up of solutions to the DGH equation,” Journal of Functional Analysis, vol. 250, no. 1, pp. 227–248, 2007.

Y. Zhou and Z. Guo, “Blow up and propagation speed of solutions to the DGH equation,” Discrete and Continuous Dynamical Systems B, vol. 12, no. 3, pp. 657–670, 2009.

J.-M. Ghidaglia, “Weakly damped forced Korteweg-de Vries equations behave as a finite-dimensional dynamical system in the long time,” Journal of Differential Equations, vol. 74, no. 2, pp. 369–390, 1988.

Z. Guo, “Blow up, global existence, and infinite propagation speed for the weakly dissipative Camassa-Holm equation,” Journal of Mathematical Physics, vol. 49, no. 3, Article ID 033516, 9 pages, 2008.

Z. Guo and L. Ni, “Wave breaking for the periodic weakly dissipative Dullin-Gottwald-Holm equation,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 3, pp. 965–973, 2011.

S. Wu and Z. Yin, “Global existence and blow-up phenomena for the weakly dissipative Camassa-Holm equation,” Journal of Differential Equations, vol. 246, no. 11, pp. 4309–4321, 2009.

A. V. Busuioc and T. S. Ratiu, “The second grade fluid and averaged Euler equations with Navier-slip boundary conditions,” Nonlinearity, vol. 16, no. 3, pp. 1119–1149, 2003.