The local strong and weak solutions for a generalized Novikov equation. (English) Zbl 1243.35044

Summary: The Kato theorem for abstract differential equations is applied to establish the local well-posedness of the strong solution for a nonlinear generalized Novikov equation in space \(C([0, T), H^s(\mathbb R)) \cap C^1([0, T), H^{s-1}(\mathbb R))\) with \(s > (3/2)\). The existence of weak solutions for the equation in the lower-order Sobolev space \(H^s(\mathbb R)\) with \(1 \leq s \leq (3/2)\) is acquired.


35G25 Initial value problems for nonlinear higher-order PDEs
35A01 Existence problems for PDEs: global existence, local existence, non-existence
35A02 Uniqueness problems for PDEs: global uniqueness, local uniqueness, non-uniqueness
35D30 Weak solutions to PDEs
Full Text: DOI


[1] V. Novikov, “Generalizations of the Camassa-Holm equation,” Journal of Physics A, vol. 42, no. 34, pp. 342002-342014, 2009. · Zbl 1181.37100
[2] A. N. W. Hone and J. P. Wang, “Integrable peakon equations with cubic nonlinearity,” Journal of Physics A, vol. 41, no. 37, pp. 372002-372010, 2008. · Zbl 1153.35075
[3] 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. · Zbl 1179.37092
[4] 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. · Zbl 1169.76010
[5] A. Constantin and J. Escher, “Particle trajectories in solitary water waves,” Bulletin of the American Mathematical Society, vol. 44, no. 3, pp. 423-431, 2007. · Zbl 1126.76012
[6] Z. Guo and Y. Zhou, “Wave breaking and persistence properties for the dispersive rod equation,” SIAM Journal on Mathematical Analysis, vol. 40, no. 6, pp. 2567-2580, 2009. · Zbl 1177.30024
[7] S. Lai and Y. Wu, “The local well-posedness and existence of weak solutions for a generalized Camassa-Holm equation,” Journal of Differential Equations, vol. 248, no. 8, pp. 2038-2063, 2010. · Zbl 1187.35179
[8] 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. · Zbl 0958.35119
[9] Y. Zhou, “Blow-up of solutions to the DGH equation,” Journal of Functional Analysis, vol. 250, no. 1, pp. 227-248, 2007. · Zbl 1124.35079
[10] Y. Zhou, “Blow-up of solutions to a nonlinear dispersive rod equation,” Calculus of Variations and Partial Differential Equations, vol. 25, no. 1, pp. 63-77, 2006. · Zbl 1172.35504
[11] 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. · Zbl 1215.37051
[12] F. Tiglay, “The periodic Cauchy problem for Novikov’s equation,” http://arxiv.org/abs/1009.1820/.
[13] 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. · Zbl 1195.35072
[14] T. Kato, “Quasi-linear equations of evolution, with applications to partial differential equations,” in Spectral Theory and Differential Equations, Lecture notes in Mathematics, Springer, Berlin, Germany, 1975. · Zbl 0315.35077
[15] T. Kato and G. Ponce, “Commutator estimates and the Euler and Navier-Stokes equations,” Communications on Pure and Applied Mathematics, vol. 41, no. 7, pp. 891-907, 1988. · Zbl 0671.35066
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.