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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 840919, 23 pages
The Cauchy Problem to a Shallow Water Wave Equation with a Weakly Dissipative Term
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.
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.
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 , 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  and Wu and Yin  discussed the weakly dissipative C-H equation The global existence of strong solutions and blow-up in finite time were presented in  provided that changes sign. The sufficient conditions on the infinite propagation speed for (1.2) are offered in . 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 ).
Degasperis and Procesi  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 ). In fact, many researchers have paid their attention to the study of solutions to (1.3). Constantin et al.  developed an inverse scattering approach for smooth localized solutions to (1.3). Liu and Yin  and Yin [21, 22] investigated the global existence of strong solutions and global weak solutions to (1.3). Henry  showed that the smooth solutions to (1.3) have infinite speed of propagation. Coclite and Karlsen  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 . The blow-up solution modeling wave breaking and the decay of solution were discussed in . The existence and uniqueness of the global weak solution in space were proved (see ).
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 .
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  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  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  and Yin .
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
Throughout this paper, let denote the mollifiers where is defined by Thus, we get Next, we give some useful results.
Lemma 3.1 (see ). Let be uniformly continuous and bounded. If , then
Lemma 3.2 (see ). Let be uniformly continuous and bounded. If , then
Lemma 3.3 (see ). Let . If and , then are a.e. equal to a function continuous from into and for all .
Lemma 3.4 (see ). 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 .
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 .
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 , 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