Abstract
In this paper we focus on the large time behaviour of momentum support for a Novikov type equation. It is shown that the momentum support can be large enough as time evolves if the initial data which is compactly supported keeps its sign.
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Anco, S.C., da Silva, P.L., Freire, I.L.: A family of wave-breaking equations generalizing the Camassa-Holm and Novikov equations. J. Math. Phys. 56, 091506 (2015)
Camassa, R., Holm, D.: An integrable shallow water equation with peaked solitons. Phys. Rev. Lett. 71, 1661–1664 (1993)
Constantin, A., Lannes, D.: The hydro-dynamical relevance of the Camassa-Holm and Degasperis-Procesi equations. Arch. Ration. Mech. Anal. 193, 165–186 (2009)
Chen, H., Guo, Z.: Asymptotic profile of solutions to the Degasperis-Procesi equation. Bull. Malays. Math. Sci. Soc. 38, 333–344 (2015)
Degasperis, A., Procesi, M.: . In: Degasperis, A., Gaeta, G. (eds.) Symmetry and Perturbation Theory, SPT 98, p 23. World Scientific, River Edge (1999)
Grayshan, K.: Peakon solutions of the Novikov equation and the properities of the date-to-solution map. J. Math. Anal. Appl. 397, 515–521 (2013)
Guo, Z.: On an integrable Camassa-Holm type equation with cubic nonlinearity. Nonlinear Anal. Real world Appl. 34, 225–232 (2017)
Guo, Z., Zhou, Y.: On solutions to a two-component generalized Camassa-Holm system. Stud. Appl. Math. 124, 307–322 (2010)
Guo, Z.: Blow-up and global solutions to a new integrable model with two components. J. Math. Anal. Appl. 372, 316–327 (2010)
Guo, Z., Li, K., Xu, C.: On a generalized Camassa-Holm type equation with (k + 1)-degree nonlinearities. Z. Angew. Math. Mech. 98, 1567–1573 (2018)
Guo, Z., Li, X., Yu, C.: Some properties of solutions to the Camassa-Holm-type equations with higher-order nonlinearities. J. Nonlinear Sci. 28, 1901–1914 (2018)
Guo, Z., Wang, W., Xu, C.: On the Camassa-Holm system with one mean zero component. Commun. Math. Sci. 14, 517–534 (2016)
Guo, Z., Zhu, M.: Wave breaking for a modified two-component Camassa-Holm system. J. Differ Equ. 252, 2759–2770 (2012)
Guo, Z., Zhu, M.: Wave breaking and measure of momentum support for an integrable Camassa-Holm system with two components. Stud. Appl. Math. 130, 417–430 (2013)
Himonas, A., Holliman, C.: The Cauchy problem for the Novikov equation. Nonlinearity 25, 449–479 (2012)
Hone, A.N.W., Lundmark, H., Szmigielski, J.: Explicit multipeakon solutions of Novikov’s cubically nonlinear integrable Camassa-Holm equation. Dyn. Partial Diff. Equ. 6, 253–289 (2009)
Hone, A.N.W., Wang, J.P.: Integrable peakon equations with cubic nonlinearity. J. Phys. A: Math. Theor. 41, 372002 (10pp) (2008)
Jiang, Z., Ni, L.: Blow-up phenomena for the integrable Novikov equation. J. Math. Anal. Appl. 385, 551–558 (2012)
Jiang, Z., Ni, L., Zhou, Y.: Wave breaking for the Camassa-Holm equation. J. Nonlinear Sci. 22, 235–245 (2012)
Jiang, Z., Zhou, Y., Zhu, M.: Large time behavior for the support of momentum density of the Camassa-Holm equation. J. Math. Phys. 54, 081503 (2013)
Kato, T., Ponce, G.: Commutator estimates and the Euler and Navier-Stokes equations. Commun. Pure. Appl. Math. 41, 891–907 (1998)
Kang, S., Tang, T.: The support of the momentum density of the Camassa-Holm equation. Appl. Math. Lett. 24, 2128–2132 (2011)
Kato, T.: Quasi-linear equations of evolution with applications to partial differential equations. In: Spectral Theory and Differential Equations. Lecture Notes in Math. vol. 448, pp. 25–70. Springer, Berlin (1975)
Lai, S.: Global weak solutions to the Novikov equation. J. Funct. Anal. 265, 520–544 (2013)
Lai, S., Wu, M.: The local strong and weak solutions to a genenralized Novikov equation. Bound. Value Probl. 2013, 134 (2013)
Lai, S., Li, N., Wu, Y.: The existence of global strong and weak solutions for the Novikov equation. J. Math. Anal. Appl. 399(2), 682–691 (2013)
Li, K., Shan, M., Xu, C., Guo, Z.: The Cauchy problem on a generalized Novikov equation. Bull. Malays. Math. Sci. Soc. 41, 1859–1877 (2018)
Mi, Y., Mu, C.: On the Cauchy problem for the modified Novikov equation with peakon solutions. J. Differ. Equ. 254, 961–982 (2013)
Novikov, V.S.: Generalizations of the Camassa-Holm equation. J. Phys. A 42, 342002 (2009)
Ni, L., Zhou, Y.: Well-posedeness and persistence properties for the Novikov equation. J. Differ. Equ. 250, 3002–3021 (2011)
Rudin, W.: Real and Complex Analysis. McGraw-Hill, New York (1987)
Rodríguez-Blanco, G.: On the Cauchy problem for the Camassa-Holm equation. Nonlinear Anal. 46, 309–327 (2001)
Tığlay, F.: The periodic Cauchy problem for Novikov’s equation. Int. Math. Res. Not. 20, 4633–4648 (2011)
Wu, X., Yin, Z.: Global weak solutions for the Novikov equation. J. Phys. A: Math. Theor. 44, 055202 (17pp) (2011)
Wu, X., Yin, Z.: Well-posedness and global existence for the Novikov equation. Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) XI, 707–727 (2012)
Yin, Z.: On the Cauchy problem for an integrable equation with peakon solutions. Illinois J. Math. 47, 649–666 (2003)
Yan, W., Li, Y., Zhang, Y.: The Cauchy problem for the integrable Novikov equation. J. Differ. Equ. 253, 298–318 (2012)
Yan, W., Li, Y., Zhang, Y.: The Cauchy problem for the Novikov equation. Nonlinear Differ. Equ. Appl. 20, 1157–1169 (2013)
Zhou, S., Chen, R.: A few remarks on the generalized Novikov equation. J. Ineq. Appl. 2013, 560 (2013)
Acknowledgments
The authors thank the referee for his/her valuable comments on the initial manuscript and also thank Professor Shaoyong Lai from Southwestern University of Finance and Economics for his fruitful discussion. This work was partially supported by the National Natural Science Foundation of China, under Grant Nos. 11301394, 11661070 and 41372264.
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Ma, C., Cao, Y. & Guo, Z. Large Time Behavior of Momentum Support for a Novikov Type Equation. Math Phys Anal Geom 22, 23 (2019). https://doi.org/10.1007/s11040-019-9317-5
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DOI: https://doi.org/10.1007/s11040-019-9317-5