- About this Journal ·
- Abstracting and Indexing ·
- Advance Access ·
- Aims and Scope ·
- Annual Issues ·
- Article Processing Charges ·
- Articles in Press ·
- Author Guidelines ·
- Bibliographic Information ·
- Citations to this Journal ·
- Contact Information ·
- Editorial Board ·
- Editorial Workflow ·
- Free eTOC Alerts ·
- Publication Ethics ·
- Reviewers Acknowledgment ·
- Submit a Manuscript ·
- Subscription Information ·
- Table of Contents
Abstract and Applied Analysis
Volume 2014 (2014), Article ID 808214, 5 pages
On the Study of Global Solutions for a Nonlinear Equation
1Department of Applied Mathematics, Southwestern University of Finance and Economics, Chengdu 610074, China
2Department of Mathematics, Xinjiang University of Finance and Economics, Urumqi 830012, China
Received 3 February 2014; Accepted 27 March 2014; Published 14 April 2014
Academic Editor: Yonghong Wu
Copyright © 2014 Haibo Yan and Ls Yong. 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.
The well-posedness of global strong solutions for a nonlinear partial differential equation including the Novikov equation is established provided that its initial value satisfies a sign condition and with . If the initial value and the mean function of satisfies the sign condition, it is proved that there exists at least one global weak solution to the equation in the space in the sense of distribution and .
Recently, Wu  obtained the existence of local solutions in the space with for the following nonlinear equation: where is a natural number, , and is a constant. Letting and , (1) becomes the Camassa-Holm equation . If , and , (1) reduces to the Novikov equation .
A lot of works have been carried out to study various dynamic properties for the Camassa-Holm and the Novikov equations. Xin and Zhang  proved that there exists a global weak solution for the Camassa-Holm equation in the space without the assumption of sign conditions on the initial value. Coclite et al.  investigated the global weak solutions for a generalized hyperelastic rod wave equation or a generalized Camassa-Holm equation. It is shown in Constantin and Escher  that the blowup occurs in the form of breaking waves; namely, the solution remains bounded but its slope becomes unbounded in finite time. After wave breaking, the solution can be continued uniquely either as a global conservative weak solution  or a global dissipative solution [8–10]. The periodic and the nonperiodic Cauchy problems for the Novikov equation were discussed by Grayshan  in the Sobolev space. Using the Galerkin-type approximation method, Himonas and Holliman  established the well-posedness for the Novikov model in the Sobolev space with on both the line and the circle. The scattering theory was employed in Hone et al.  to find nonsmooth explicit soliton solutions with multiple peaks for the Novikov equation. Wu and Zhong  proved the existence of local strong and weak solutions for a generalized Novikov equation.
The objective of this work is to study (1) with . Namely, we investigate the problem where , and are described in (1). Assuming that the initial value satisfies a sign condition and , we will show that there exists a unique global strong solution in the Sobolev space . If the initial value and the mean function of satisfies the sign condition, it is shown that there exists at least one global weak solution to the equation in the space in the sense of distribution and .
2. Main Results
We define and let with . For the convolution , we know that for any with . Notation (or equivalently ) means that the mean function of is nonnegative; namely, (or equivalently ) for an arbitrary sufficiently small . For and nonnegative number , we let denote the Frechet space of all continuous -valued functions on and write .
We state the result of global strong solutions for problem (2).
Theorem 1. Let , and for all or for all . Then problem (2) has a unique strong solution satisfying
Definition 2. A function is called a global weak solution to problem (2) if for every and all , it holds that with .
Now we give the main result of global weak solution for problem (2).
Theorem 3. Let , (or equivalently ). Then problem (2) has a unique global weak solution in the sense of distribution and .
3. Several Lemmas
Using the first equation of system (2) derives which yields the conservation law
Lemma 5 (see ). Let and the function is a solution of problem (2) and the initial data . Then the following inequalities hold:
For , there is a constant such that
For , there is a constant such that
Consider the differential equation where is the solution of problem (2) and is the maximal existence time of the solution.
Proof. From Lemma 4, we know that there exists a unique solution
The Sobolev imbedding theorem derives . This means that two functions and are bounded, Lipschitz in space and in time. Using the existence and uniqueness theorem of ordinary differential equations, we derive that problem (12) has a unique solution .
Differentiating (12) with respect to gives rise to from which we obtain For every , applying the Sobolev imbedding theorem results in
Therefore, we know that there exists a constant such that for . The proof is completed.
Lemma 7. Let with , and let be the maximal existence time of the problem (2); it holds that where and .
Proof. We have from which we have Using completes the proof.
Lemma 8. If , or , then the solution of problem (2) satisfies
Proof. We only need to prove this lemma for the case since the proof of the other case is similar. It follows from Lemmas 6 and 7 that . Letting , we have which derives On the other hand, we have The inequalities (22) and (23) derive that inequality (20) is valid.
Lemma 9. For , and , it holds that where is a constant independent of .
Lemma 10. Provided that , and (or equivalently ), then there exists a constant independent of and such that the solution of problem (25) satisfies
4. Proof of Main Results
Provided that , for problem (25), applying Lemmas 5, 8, and 10, and the Gronwall’s inequality, we obtain the inequalities where , and is a constant independent of and . Using the Aubin compactness theorem, we know that that there is a subsequence of such that and their temporal derivatives converge weakly to a function and its derivative in the space and , respectively, where is an arbitrary fixed positive number. In addition, for any real number , converges strongly to the function in the space for and converges strongly to in the space for .
Proof of Theorem 3. For an arbitrary fixed , using Lemma 10, we know that is bounded in the space . Therefore, we derive that the sequences , , , and converge weakly to , , , and in for any , separately. Applying the identity , we conclude that satisfies the equation where . We know that is a separable Banach space and is a bounded sequence in the dual space of . Thus, there exists a subsequence of , still denoted by , weakly star convergent to a function in . Since weakly converges to in , it derives that almost everywhere. Therefore, we obtain . Since is an arbitrary number, we complete the proof of existence of global weak solutions to problem (2).
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
This work is supported by the Fundamental Research Funds for the Central Universities (JBK120504).
- M. Wu, “On the study of local solutions for a generalized Camassa-Holm equation,” Abstract and Applied Analysis, vol. 2012, Article ID 164876, 17 pages, 2012.
- R. Camassa and D. D. Holm, “An integrable shallow water equation with peaked solitons,” Physical Review Letters, vol. 3, no. 11, pp. 1661–1664, 1993.
- V. Novikov, “Generalizations of the Camassa-Holm equation,” Journal of Physics A: Mathematical and Theoretical, vol. 42, no. 34, Article ID 342002, 2009.
- 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.
- G. M. Coclite, H. Holden, and K. H. Karlsen, “Global weak solutions to a generalized hyperelastic-rod wave equation,” SIAM Journal on Mathematical Analysis, vol. 37, no. 4, pp. 1044–1069, 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. Bressan and A. Constantin, “Global conservative solutions of the Camassa-Holm equation,” Archive for Rational Mechanics and Analysis, vol. 183, no. 1, pp. 215–239, 2007.
- A. Bressan and A. Constantin, “Global dissipative solutions of the Camassa-Holm equation,” Analysis and Applications, vol. 5, no. 1, pp. 1–27, 2007.
- H. Holden and X. Raynaud, “Dissipative solutions for the Camassa-Holm equation,” Discrete and Continuous Dynamical Systems A, vol. 24, no. 4, pp. 1047–1112, 2009.
- H. Holden and X. Raynaud, “Global conservative solutions of the Camassa-Holm equation—a lagrangian point of view,” Communications in Partial Differential Equations, vol. 32, no. 10–12, pp. 1511–1549, 2007.
- K. Grayshan, “Peakon solutions of the Novikov equation and properties of the data-to-solution map,” Journal of Mathematical Analysis and Applications, vol. 397, no. 2, pp. 515–521, 2013.
- A. A. Himonas and C. Holliman, “The Cauchy problem for the Novikov equation,” Nonlinearity, vol. 25, no. 2, pp. 449–479, 2012.
- A. N. W. Hone, H. Lundmark, and J. Szmigielski, “Explicit multipeakon solutions of Novikov's cubically nonlinear integrable Camassa-Holm type equation,” Dynamics of Partial Differential Equations, vol. 6, no. 3, pp. 253–289, 2009.
- M. Wu and Y. Zhong, “The local strong and weak solutions for a generalized Novikov equation,” Abstract and Applied Analysis, vol. 2012, Article ID 158126, 14 pages, 2012.
- S. Lai and Y. Wu, “A model containing both the Camassa-Holm and Degasperis-Procesi equations,” Journal of Mathematical Analysis and Applications, vol. 374, no. 2, pp. 458–469, 2011.
- N. Li, S. Lai, S. Li, and M. Wu, “The local and global existence of solutions for a generalized Camassa-Holm equation,” Abstract and Applied Analysis, vol. 2012, Article ID 532369, 26 pages, 2012.