The regularity of local solutions for a generalized Camassa-Holm type equation. (English) Zbl 1184.35094
Summary: The regularity of the Cauchy problem for a generalized Camassa-Holm type equation is investigated. The pseudoparabolic regularization approach is employed to obtain some priori estimates under certain assumptions on the initial value of the equation. The local existence of its solution in the Sobolev space \(H^s (\mathbb R)\) with \(1 < s \leq \frac{3}{2}\) is derived.
MSC:
| 35B65 | Smoothness and regularity of solutions to PDEs |
| 35G25 | Initial value problems for nonlinear higher-order PDEs |
| 35B45 | A priori estimates in context of PDEs |
| 35K70 | Ultraparabolic equations, pseudoparabolic equations, etc. |
References:
| [1] | Camassa, R., Holm, D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett., 71, 1661–1664 (1993) · doi:10.1103/PhysRevLett.71.1661 |
| [2] | Camassa, R., Holm, D., Hyman, J. M.: A new integral shallow water equation. Adv. Appl. Mech., 31, 1–33 (1994) · Zbl 0808.76011 · doi:10.1016/S0065-2156(08)70254-0 |
| [3] | Li, Y., Olver, P.: Well-posedness and blow solutions for an integrable nonlinearly dispersive model wave equation. J. Diff. Eqs., 162, 27–63 (2000) · Zbl 0958.35119 · doi:10.1006/jdeq.1999.3683 |
| [4] | Xin, Z., Zhang, P.: On the weak solutions to a shallow water equation. Comm. Pure Appl. Math., 53, 1411–1433 (2000) · Zbl 1048.35092 · doi:10.1002/1097-0312(200011)53:11<1411::AID-CPA4>3.0.CO;2-5 |
| [5] | Xin, X., Zhang, P.: On the uniqueness and large time behavior of the weak solutions to a shallow water equation. Comm. Partial Differential Equations, 27(9–10), 1815–1844 (2002) · Zbl 1034.35115 · doi:10.1081/PDE-120016129 |
| [6] | Wazwaz, A. M.: A class of nonlinear fourth order variant of a generalized Camassa-Holm equation with compact and noncompact solutions. Appl. Math. Comput., 165, 485–501 (2005) · Zbl 1070.35044 · doi:10.1016/j.amc.2004.04.029 |
| [7] | Lim, W. K.: Global well-posedness for the viscous Camassa-Holm equation. J. Math. Anal. Appl., 326, 432–442 (2007) · Zbl 1160.35064 · doi:10.1016/j.jmaa.2006.01.095 |
| [8] | Tian, L. X., Song X. Y.: New peaked solitary wave solutions of the generalized Camassa-Holm equation. Chaos, Solitons and Fractals, 19, 621–637 (2004) · Zbl 1068.35123 · doi:10.1016/S0960-0779(03)00192-9 |
| [9] | Yin, Z. Y.: On the blow-up scenario for the generalized Camassa-Holm equation. Comm. Partial Differential Equations, 29, 867–877 (2004) · Zbl 1068.35030 · doi:10.1081/PDE-120037334 |
| [10] | Fu, Y. P., Guo, B. L.: Time periodic solution of the viscous Camassa-Holm equation. J. Math. Anal. Appl., 313, 311–321 (2006) · Zbl 1078.35012 · doi:10.1016/j.jmaa.2005.08.073 |
| [11] | Ma, L., Chen, D. Z., Yang, Y.: Some results on subelliptic equations. Acta Mathematica Sinica, English Series, 22, 1695–1704 (2006) · Zbl 1134.35055 · doi:10.1007/s10114-005-0882-0 |
| [12] | Miao, C. X., Zhu, Y. B.: Well-posedness in the energy space for non-linear system of wave equations with critical growth. Acta Mathematica Sinica, English Series, 23, 17–26 (2007) · Zbl 1101.11025 · doi:10.1007/s10114-005-0766-3 |
| [13] | Kokilashili, V., S. Samko, S.: Boundedness of maximal operators and potential operators on Carleson curves in Lebesgue spaces with variable exponent. Acta Mathematica Sinica, English Series, 24, 1775–1800 (2008) · Zbl 1151.42006 · doi:10.1007/s10114-008-6464-1 |
| [14] | Ji, R. H., Zheng, S. N.: Asymptotic estimates to non-global solutions of a multi-coupled parabolic system. Acta Mathematica Sinica, English Series, 24, 1713–1726 (2008) · Zbl 1180.35282 · doi:10.1007/s10114-008-6559-8 |
| [15] | Bona, J., Smith, R.: The initial value problem for the Korteweg-de Vries equation. Phil. Trans. Roy. Soc. London Ser. A, 278, 555–601 (1975) · Zbl 0306.35027 · doi:10.1098/rsta.1975.0035 |
| [16] | Kato, T., Ponce, G.: Commutator estimayes and the Euler and Navier-Stokes equation. Commun. Pure Appl. Math., 41, 891–907 (1988) · Zbl 0671.35066 · doi:10.1002/cpa.3160410704 |
| [17] | Walter, W.: Differential and Integral Inequalities, Springer-Verlag, New York, 1970 · Zbl 0252.35005 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
