- About this Journal ·
- Abstracting and Indexing ·
- 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 2013 (2013), Article ID 619068, 5 pages
The Local Strong Solutions and Global Weak Solutions for a Nonlinear Equation
Department of Mathematics, Southwestern University of Finance and Economics, Chengdu 610074, China
Received 31 March 2013; Accepted 26 April 2013
Academic Editor: Shaoyong Lai
Copyright © 2013 Meng Wu. 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 existence and uniqueness of local strong solutions for a nonlinear equation are investigated in the Sobolev space provided that the initial value lies in with . Meanwhile, we prove the existence of global weak solutions in for the equation.
Coclite and Karlsen  investigated the well posedness in classes of discontinuous functions for the generalized Degasperis-Procesi equation: which is subject to the condition or where is a positive constant. The existence and stability of entropy weak solutions belonging to the class are established for (1) in paper .
In this work, we study the following model: where is a positive constant and . If and , (4) reduces to the classical Degasperis-Procesi model (see [2–13]). Here, we notice that assumptions (2) and (3) do not include the case . In this paper, we will study the case , and is an arbitrary positive constant.
It is shown in this work that there exists a unique local strong solution in the Sobolev space by assuming that the initial value belongs to with . In addition, we prove the existence of global weak solutions in for the system (6).
2. Local Existence
In this section, we will use the Kato theorem in  for abstract differential equation to establish the existence of local strong solution for the problem (6). Let us consider the following problem: Let and be Hilbert spaces such that is continuously and densely embedded in , and let be a topological isomorphism. Let be the space of all bounded linear operators from to . In the case of , we denote this space by . We illustrate the following conditions in which , , , and are constants depending only on .(i) for with and (i.e., is quasi-m-accretive), uniformly on bounded sets in .(ii), where is bounded, uniformly on bounded sets in . Moreover, (iii) extends to a map from into , is bounded on bounded sets in , and satisfies
Kato Theorem (see ). Assume that conditions (i), (ii), and (iii) hold. If , there is a maximal depending only on and a unique solution to the problem (7) such that Moreover, the map is a continuous map from to the following space:
In order to apply the Kato theorem to establish the local well posedness for the problem (6), we let , , , , , and . We know that is an isomorphism of onto . Now, we cite the following Lemmas.
Lemma 1. The operator with , belongs to .
Lemma 2. Assume that with and . Then, for all . Moreover,
Lemma 3. For , and , it holds that for and
The above three Lemmas can be found in Ni and Zhou .
Lemma 4. Let with and . Then, is bounded on bounded sets in and satisfies
Proof. For , we know that . Consequently, we have
3. Weak Solutions
Lemma 6. The solution of the problem (5) with satisfies where , and . Moreover, there exist two constants and depending only on such that
Lemma 7. Let , , and let be the maximal existence time of the solution to the problem (6). Then, the problem (23) has a unique solution . Moreover, the map is an increasing diffeomorphism of with for .
Proof. Using Theorem 5, we obtain and . Therefore, we know that functions and are bounded, Lipschitz in space, and in time. Using the existence and uniqueness theorem for ordinary differential equations derives that the problem (23) has a unique solution .
Differentiating (23) with respect to gives rise to the following: from which we obtain For every , using the Sobolev imbedding theorem yields that
It is inferred that there exists a constant such that for . It completes the proof.
Lemma 8. Assume that . Let be the maximal existence time of the solution to the problem (6). Then, it has where is a constant independent of .
Proof. Let , we have for all and . Using a simple density argument presented in , it suffices to consider to prove this lemma. Let be the maximal existence time of the solution to the problem (6) with the initial value such that . From (6), we have Since from (29), we have Using Lemma 6 and (30) derives that where is a positive constant independent of . Using (31) results in the following: Therefore, Using the Sobolev embedding theorem to ensure the uniform boundedness of for with , from Lemma 7, for every , we get a constant such that We deduce from (34) that the function is strictly increasing on with as long as . It follows from (33) that Using the Gronwall inequality and (35) derives that (27) holds.
For a real number with , suppose that the function is in , and let be the convolution of the function and such that the Fourier transform of satisfies , , and for any . Then, we have . It follows from Theorem 5 that for each satisfying , the Cauchy problem, has a unique solution . Using Lemmas 6 and 8, for every , we obtain Sending , we know that inequalities (37) are still valid. This means that for , (37) hold.
Now, we state the concepts of weak solutions.
Definition 9 (weak solution). We call a function a weak solution of the Cauchy problem (5) provided that(i);(ii) in , that is, for all there holds the following identity:
Theorem 10. Let . Then, there exists a weak solution to the problem (5).
Proof. Consider the problem (36). For an arbitrary , choosing a subsequence , from (37), we know that is bounded in and is uniformly bounded in . Therefore, we obtain that is bounded in . Therefore, there exist subsequences and , still denoted by and , are weakly convergent to in . Noticing (38) completes the proof.
This work is supported by the Fundamental Research Funds for the Central Universities (JBK120504).
- 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.
- A. Constantin and D. Lannes, “The hydrodynamical relevance of the Camassa-Holm and Degasperis-Procesi equations,” Archive for Rational Mechanics and Analysis, vol. 192, no. 1, pp. 165–186, 2009.
- J. Escher, Y. Liu, and Z. Y. 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.
- Y. Fu, Y. Liu, and C. Z. Qu, “On the blow-up structure for the generalized periodic Camassa-Holm and Degasperis-Procesi equations,” Journal of Functional Analysis, vol. 262, no. 7, pp. 3125–3158, 2012.
- 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. L. Gui, Y. Liu, and L. X. Tian, “Global existence and blow-up phenomena for the peakon -family of equations,” Indiana University Mathematics Journal, vol. 57, no. 3, pp. 1209–1234, 2008.
- Y. Liu and Z. Y. Yin, “Global existence and blow-up phenomena for the Degasperis-Procesi equation,” Communications in Mathematical Physics, vol. 267, no. 3, pp. 801–820, 2006.
- S. Y. Lai and Y. H. 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.
- S. Y. Lai and A. Y. Wang, “The well-posedness of solutions for a generalized shallow water wave equation,” Abstract and Applied Analysis, vol. 2012, Article ID 872187, 15 pages, 2012.
- S. Y. Lai, “The global weak solutions for a generalized Camassa-Holm equation,” Abstract and Applied Analysis, vol. 2013, Article ID 838302, 6 pages, 2013.
- Z. W. Lin and Y. Liu, “Stability of peakons for the Degasperis-Procesi equation,” Communications on Pure and Applied Mathematics, vol. 62, no. 1, pp. 125–146, 2009.
- O. G. Mustafa, “A note on the Degasperis-Procesi equation,” Journal of Nonlinear Mathematical Physics, vol. 12, no. 1, pp. 10–14, 2005.
- Z. Y. 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.
- T. Kato, “Quasi-linear equations of evolution, with applications to partial differential equations,” in Spectral Theory and Differential Equations, Lecture Notes in Mathematics, pp. 25–70, Springer, Berlin, Germany, 1975.
- L. Ni and Y. Zhou, “Well-posedness and persistence properties for the Novikov equation,” Journal of Differential Equations, vol. 250, no. 7, pp. 3002–3021, 2011.