Abstract and Applied AnalysisVolume 2012 (2012), Article ID 840919, 23 pagesdoi:10.1155/2012/840919

Research Article

The Cauchy Problem to a Shallow Water Wave Equation with a Weakly Dissipative Term

Ying Wang and YunXi Guo

College of Science, Sichuan University of Science and Engineering, Zigong 643000, China

Received 21 February 2012; Accepted 11 March 2012

Academic Editor: Shaoyong Lai

Copyright © 2012 Ying Wang and YunXi Guo. 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.

A shallow water wave equation with a weakly dissipative term, which includes the weakly dissipative Camassa-Holm and the weakly dissipative Degasperis-Procesi equations as special cases, is investigated. The sufficient conditions about the existence of the global strong solution are given. Provided that , and , the existence and uniqueness of the global weak solution to the equation are shown to be true.

1. Introduction

The Camassa-Holm equation (C-H equation) as a model for wave motion on shallow water, has a bi-Hamiltonian structure and is completely integrable. After the equation was derived by Camassa and Holm [1], a lot of works was devoted to its investigation of dynamical properties. The local well posedness of solution for initial data with was given by several authors [2–4]. Under certain assumptions, (1.1) has not only global strong solutions and blow-up solutions [2, 5–7] but also global weak solutions in (see [8–10]). For other methods to handle the problems related to the Camassa-Holm equation and functional spaces, the reader is referred to [11–14] and the references therein.

To study the effect of the weakly dissipative term on the C-H equation, Guo [15] and Wu and Yin [16] discussed the weakly dissipative C-H equation The global existence of strong solutions and blow-up in finite time were presented in [16] provided that changes sign. The sufficient conditions on the infinite propagation speed for (1.2) are offered in [15]. It is found that the local well posedness and the blow-up phenomena of (1.2) are similar to the C-H equation in a finite interval of time. However, there are differences between (1.2) and the C-H equation in their long time behaviors. For example, the global strong solutions of (1.2) decay to zero as time tends to infinite under suitable assumptions, which implies that (1.2) has no traveling wave solutions (see [16]).

Degasperis and Procesi [17] derived the equation (D-P equation) as a model for shallow water dynamics. Although the D-P equation (1.3) has a similar form to the C-H equation (1.1), it should be addressed that they are truly different (see [18]). In fact, many researchers have paid their attention to the study of solutions to (1.3). Constantin et al. [19] developed an inverse scattering approach for smooth localized solutions to (1.3). Liu and Yin [20] and Yin [21, 22] investigated the global existence of strong solutions and global weak solutions to (1.3). Henry [23] showed that the smooth solutions to (1.3) have infinite speed of propagation. Coclite and Karlsen [24] obtained global existence results for entropy solutions in and in .

The weakly dissipative D-P equation is investigated by several authors [25–27] to find out the effect of the weakly dissipative term on the D-P equation. The global existence, persistence properties, unique continuation and the infinite propagation speed of the strong solutions to (1.4) are studied in [26]. The blow-up solution modeling wave breaking and the decay of solution were discussed in [27]. The existence and uniqueness of the global weak solution in space were proved (see [25]).

In this paper, we will consider the Cauchy problem for the weakly dissipative shallow water wave equation where , and are arbitrary constants, is the fluid velocity in the direction, represents the weakly dissipative term. For , we notice that (1.5) is a special case of the shallow water equation derived by Constantin and Lannes [28].

Since (1.5) is a generalization of the Camassa-Holm equation and the Degasperis-Procesi equation, (1.5) loses some important conservation laws that C-H equation and D-P equation possesses. It needs to be pointed out that Lai and Wu [12] studied global existence and blow-up criteria for (1.5) with . To the best of our knowledge, the dynamical behaviors related to (1.5) with , such as the global weak solution in space , have not been yet investigated. The objective of this paper is to investigate several dynamical properties of solutions to (1.5). More precisely, we firstly use the Kato theorem [29] to establish the local well-posedness for (1.5) with initial value with . Then, we present a precise blow-up scenario for (1.5). Provided that and the potential does not change sign, the global existence of the strong solution is shown to be true. Finally, under suitable assumptions, the existence and uniqueness of global weak solution in are proved. Our main ideas to prove the existence and uniqueness of the global weak solution come from those presented in Constantin and Molinet [8] and Yin [22].

2. Notations

The space of all infinitely differentiable functions with compact support in is denoted by . Let , and let be the space of all measurable functions such that . We define with the standard norm . For any real number , let denote the Sobolev space with the norm defined by , where .

We denote by the convolution. Let denote the norm of Banach space and the , duality bracket. Let be the space of the Radon measures on R with bounded total variation and the subset of positive measures. Finally, we write for the space of functions with bounded variation, being the total variation of .

Note that if , . Then, for all and . Using this identity, we rewrite problem (1.5) in the form which is equivalent to

3. Preliminaries

Throughout this paper, let denote the mollifiers where is defined by Thus, we get Next, we give some useful results.

Lemma 3.1 (see [8]). Let be uniformly continuous and bounded. If , then

Lemma 3.2 (see [8]). Let be uniformly continuous and bounded. If , then

Lemma 3.3 (see [30]). Let . If and , then are a.e. equal to a function continuous from into and for all .

Lemma 3.4 (see [8]). Assume that is uniformly bounded in for all . Then, for a.e.

4. Global Strong Solution

We firstly present the existence and uniqueness of the local solutions to the problem (2.1).

Theorem 4.1. Let . Then, the problem (2.1) has a unique solution , such that where depends on .

Proof. The proof of Theorem 4.1 can be finished by using Kato’s semigroup theory (see [29] or [4]). Here, we omit the detailed proof.

Theorem 4.2. Given , the solution of problem (2.1) blows up in finite time if and only if

Proof. Setting , we get which yields Using system (2.2), one has Assume that there is a constant such that From (4.5), we get Using Gronwall’ inequality, we deduce the is bounded on . On the other hand, Therefore, using (4.4) leads to It shows that if is bounded, then is also bounded. This completes the proof.

We consider the differential equation where solves (1.5) and .

Lemma 4.3. Let ; is the maximal existence time of the corresponding solution to (1.5). Then, system (4.10) has a unique solution . Moreover, the map is an increasing diffeomorphism of with

Proof. From Theorem 4.1, we have and . We conclude that both functions and are bounded, Lipschitz in space, and in time. Applying the existence and uniqueness theorem of ordinary differential equations implies that system (4.10) has a unique solution .Differentiating (4.10) with respect to leads to which yields For every , using the Sobolev embedding theorem gives rise to It is inferred that there exists a constant such that for .By computing directly, we derive which results in The proof of Lemma 4.3 is completed.

Theorem 4.4. Let , and for all (or equivalently for all ). Then, problem (2.1) has a global strong solution Moreover, if , then one has for all , (i), , and on R,(ii) and ,(iii).

Proof. Let , and let be the maximal existence time of the solution with initial date (cf. Theorem 4.1). If , then Theorem 4.2 ensures that for all .Note that and . By Young’s inequality, we get Integrating the first equation of problem (2.1) by parts, we get It follows that Since , we have Given , due to , from Theorem 4.1 and (4.21), we obtain Note that ,   on and the positivity of . Thus, we can infer that on . From (4.22) we have From Theorem 4.2 and (4.23), we find . This implies that problem (2.1) has a unique solution Due to and for all , it shows that From the two identities above, we infer that on for all . This proves .Due to , we obtain From and the inequality above, we get On the other hand, from (4.23), we have that . This proves .Multiplying the first equation of problem (2.1) by and integrating by parts, we find which yields From Gronwall’s inequality, one has This proves (iii) and completes the proof of the theorem.

5. Global Weak Solution

Theorem 5.1. Let and . Then equation (1.5) has a unique solution with initial data and such that is uniformly bounded on R.

Proof. We split the proof of Theorem 5.1 in two parts.Let and . Note that . Thus, for , we have Let us define for . Obviously, we get Note that, for all , Referring to the proof of (5.1), we have From the Theorem 4.4, we know that there exists a global strong solution and for all .Note that for all From Theorem 4.4 and (5.2), we obtain From the Hölder inequality, Theorem 4.4, and (5.2), for all and , we have Using Young’s inequality, we get where is bounded, and Applying (5.8)–(5.10) and problem (2.1), we have For fixed , from (5.7) and (5.11), we deduce where is a positive constant depending only on , , , and . It follows that the sequence is uniformly bounded in the space . Thus, we can extract a subsequence such that for some . From Theorem 4.4 and (5.2), for fixed , we have that the sequence satisfies Applying Helly’s theorem [31], we infer that there exists a subsequence, denoted again by , which converges at every point to some function of finite variation with From (5.14), we get that for almost all , in , it follows that for a.e. . Therefore, we have and, for a.e. , By Theorem 4.4 and (5.7), we have Note that for fixed , the sequence is uniformly bounded in . Therefore, it has a subsequence , which converges weakly in . From (5.14), we infer that the weak -limit is . It follows from that From (5.14), (5.17), and(5.20), we have that solves (2.1) in .For fixed , note that is uniformly bounded in as and is uniformly bounded for all and , and we infer that the family is weakly equicontinous on . An application of the Arzela-Ascoli theorem yields that has a subsequence, denoted again , which converges weakly in , uniformly in . The limit function is . being arbitrary, we have that is locally and weakly continuous from into , that is, .Since, for a.e. , weakly in , from Theorem 4.4, we get Inequality (5.21) shows that From Theorem 4.4, (5.1) and (5.2), for , we obtain Combining with (5.14), we have Next, we will prove that by using a regularization approach.Since satisfies (2.1) in distribution sense, convoluting (2.1) with , we have that, for a.e., Integrating the above equation with respect to on , we obtain Integration by parts gives rise to Utilizing Lemma 3.3, we obtain that Since it follows that, for a.e. , Finally, we prove that is uniformly bounded on and .Due to from (5.18), we get that, for a.e. , The above inequality implies that, for a.e. , is uniformly bounded on . For fixed , applying (5.13) and (5.14), we have Since for all , we obtain that for a.e. , .Note that . Then we get Combining with (5.24), it implies that .This completes the proof of the existence of Theorem 5.1.Next, we present the uniqueness proof of the Theorem 5.1.Let be two global weak solutions of problem (2.1) with the same initial data . Assume that and are uniformly bounded on and set From assumption, we know that . Then, for all , Similarly, Following the same procedure as in (5.1), we may also get that and, for all , We define Convoluting (2.1) for and with , we get that for a.e. and all , Subtracting (5.43) from (5.42) and using Lemma 3.4, integration by parts shows that, for a.e. and all Using (5.36)–(5.38) and Young’s inequality to the first term on the right-hand side of (5.44) yields, Similarly, for the second term and the third term on the right-hand side of (5.44), we have For the last term on the right-hand side of (5.44), we have From (5.45)–(5.47), for a.e. and all , we find where where is a positive constant depending on and the -norms of and .In the same way, convoluting (2.1) for and with and using Lemma 3.4, we get that, for a.e. and all , Using the identity for and Young’s inequality, we estimate the forth term of the right-hand side of (5.50) Using (5.36)–(5.38) and Young’s inequality to the first term on the right-hand side of (5.50) gives rise to To treat the second term of the right-hand side of (5.50), we note that Applying Lemma 3.1, the second expression of the right-hand side of (5.53) can be estimated by a function belonging to (5.49). Making use of the Hölder inequality and (5.1), for a.e. and all , we have It follows from (5.53) and (5.54) that Now, we deal with the third term on the right-hand side of (5.50) Therefore, (5.56) implies that, for a.e. and all , From (5.51), (5.52), (5.55), and (5.57), for a.e. and all , we deduce that Combining with (5.48) and (5.58), we find It follows from Gronwall’ inequality that, for a.e. and all , Fix , and let in (5.60). Since and relation (5.49) holds, making use of Lebesgue’s dominated convergence theorem yields Note that ; therefore, we obtain for a.e. . This completes the proof of the theorem.

Acknowledgments

This work was supported by RFSUSE (no. 2011KY12), RFSUSE (no. 2011RC10) and the project (no. 12ZB080). The authors thanks the referee for valuable comments.

References

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 and J. Escher, “Global existence and blow-up for a shallow water equation,” Annali della Scuola Normale Superiore di Pisa, vol. 26, no. 2, pp. 303–328, 1998.

Y. A. Li and P. J. Olver, “Well-posedness and blow-up solutions for an integrable nonlinearly dispersive model wave equation,” Journal of Differential Equations, vol. 162, no. 1, pp. 27–63, 2000.

G. Rodríguez-Blanco, “On the Cauchy problem for the Camassa-Holm equation,” Nonlinear Analysis: Theory, Methods & Applications, vol. 46, no. 3, pp. 309–327, 2001.

A. Constantin, “Existence of permanent and breaking waves for a shallow water equation: a geometric approach,” Annales de l'Institut Fourier, vol. 50, no. 2, pp. 321–362, 2000.

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 H. P. McKean, “A shallow water equation on the circle,” Communications on Pure and Applied Mathematics, vol. 52, no. 8, pp. 949–982, 1999.

A. Constantin and L. Molinet, “Global weak solutions for a shallow water equation,” Communications in Mathematical Physics, vol. 211, no. 1, pp. 45–61, 2000.

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.

Z. Xin and P. Zhang, “On the uniqueness and large time behavior of the weak solutions to a shallow water equation,” Communications in Partial Differential Equations, vol. 27, no. 9-10, pp. 1815–1844, 2002.

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.

S. Lai and Y. Wu, “Existence of weak solutions in lower order Sobolev space for a Camassa-Holm-type equation,” Journal of Physics A, vol. 43, no. 9, Article ID 095205, 13 pages, 2010.

Y. Zhou, “On solutions to the Holm-Staley b-family of equations,” Nonlinearity, vol. 23, no. 2, pp. 369–381, 2010.

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

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.

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. Degasperis and M. Procesi, “Asymptotic integrability,” in Symmetry and Perturbation Theory (Rome, 1998), pp. 23–37, World Scientific Publishing, River Edge, NJ, USA, 1999.

J. Escher, Y. Liu, and Z. Yin, “Global weak solutions and blow-up structure for the Degasperis-Procesi equation,” Journal of Functional Analysis, vol. 241, no. 2, pp. 457–485, 2006.

A. Constantin, R. I. Ivanov, and J. Lenells, “Inverse scattering transform for the Degasperis-Procesi equation,” Nonlinearity, vol. 23, no. 10, pp. 2559–2575, 2010.

Y. Liu and Z. Yin, “Global existence and blow-up phenomena for the Degasperis-Procesi equation,” Communications in Mathematical Physics, vol. 267, no. 3, pp. 801–820, 2006.

Z. Yin, “Global weak solutions for a new periodic integrable equation with peakon solutions,” Journal of Functional Analysis, vol. 212, no. 1, pp. 182–194, 2004.

Z. Yin, “Global solutions to a new integrable equation with peakons,” Indiana University Mathematics Journal, vol. 53, no. 4, pp. 1189–1209, 2004.

D. Henry, “Infinite propagation speed for the Degasperis-Procesi equation,” Journal of Mathematical Analysis and Applications, vol. 311, no. 2, pp. 755–759, 2005.

G. M. Coclite and K. H. Karlsen, “On the well-posedness of the Degasperis-Procesi equation,” Journal of Functional Analysis, vol. 233, no. 1, pp. 60–91, 2006.

Y. Guo, S. Lai, and Y. Wang, “Global weak solutions to the weakly dissipative Degasperis-Procesi equation,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 15, pp. 4961–4973, 2011.

Z. Guo, “Some properties of solutions to the weakly dissipative Degasperis-Procesi equation,” Journal of Differential Equations, vol. 246, no. 11, pp. 4332–4344, 2009.

S. Wu and Z. Yin, “Blow-up and decay of the solution of the weakly dissipative Degasperis-Procesi equation,” SIAM Journal on Mathematical Analysis, vol. 40, no. 2, pp. 475–490, 2008.

A. Constantin and D. Lannes, “The hydro-dynamical relevance of the Camassa-Holm and Degasperis-Procesi equations,” Archive for Rational Mechanics and Analysis, vol. 193, pp. 165–186, 2009.

T. Kato, “Quasi-linear equations of evolution, with applications to partial differential equations,” in Spectral Theory and Differential Equations, Lecture Notes in Mathematics Vol. 448, pp. 25–70, Springer, Berlin, Germany, 1975.

J. Málek, J. Nečas, M. Rokyta, and M. Ruzicka, Weak and Measure-valued Solutions to Evolutionary PDEs, vol. 13 of Applied Mathematics and Mathematical Computation, Chapman & Hall, London, UK, 1996.

I. P. Natanson, Theory of Functions of a Real Variable, F. Ungar Publishing Co, New York, NY, USA, 1964.