ON THE PROPAGATION OF REGULARITY OF SOLUTIONS TO THE NONLINEAR FIFTH ORDER EQUATION OF KDV TYPE

Boling Guo; Ying Zhang

doi:10.11948/20220377

2023 Volume 13 Issue 5

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2023 > 13(5): 2471-2486

Next Article Previous Article

Boling Guo, Ying Zhang. ON THE PROPAGATION OF REGULARITY OF SOLUTIONS TO THE NONLINEAR FIFTH ORDER EQUATION OF KDV TYPE[J]. Journal of Applied Analysis & Computation, 2023, 13(5): 2471-2486. doi: 10.11948/20220377

Citation:

Boling Guo, Ying Zhang. ON THE PROPAGATION OF REGULARITY OF SOLUTIONS TO THE NONLINEAR FIFTH ORDER EQUATION OF KDV TYPE[J]. Journal of Applied Analysis & Computation, 2023, 13(5): 2471-2486. doi: 10.11948/20220377

ON THE PROPAGATION OF REGULARITY OF SOLUTIONS TO THE NONLINEAR FIFTH ORDER EQUATION OF KDV TYPE

Boling Guo^1,,
Ying Zhang^2, ,

1.
Institute of Applied Physics and Computational Mathematics, Beijing, 100088, China
2.
The Graduate School of China Academy of Engineering Physics, Beijing, 100088, China

Author Bio: Email: gbl@iapcm.ac.cn(B. Guo)
Corresponding author: Email: zhangying21@gscaep.ac.cn(Y. Zhang)

Abstract

Abstract

We investigate special regularity of solutions to the initial value problem associated to the nonlinear fifth order equation of KdV type. The main results show that for datum $u_{0} \in H^{s}({\bf R}), F(u) \in C^{s+2}({\bf R})$ with $s\geq5$, whose restriction belongs to $H^{l}((x_{0}, \infty))$ and $H^{l+2}((x_{0}, \infty))$ respectively, for some $l \in \mathbb{Z}^{+}$ and $x_{0}\in {\bf R}$, then the restriction of the corresponding solution $u(\cdot, t)$ belongs to $H^{l}((b, \infty))$ for any $b\in{\bf R}$ and any $t\in (0, T)$. Thus, this type of regularity travels with infinite speed to its left as time evolves. To a certain extent, our results complement the previous studies on the related aspects, and deepen the understanding of such properties for the dispersion equation.
- Nonlinear fifth order equation /
- Sobolev space /
- propagation of regularity
MSC: 35Q53, 35B05

FullText(HTML)

References

[1]	B. Guo, Y. Han and Y. Zhou, On smooth solution for a nonlinear 5th order equation of KdV type, J. Partial Differ. Equ., 1995, 8, 321–332. Google Scholar
[2]	B. Guo and G. Qin, On the propagation of regularity and decay of solutions to the Benjamin equation, J. Math. Phys., 2018, 59(7), 071505. doi: 10.1063/1.5026916 CrossRef Google Scholar
[3]	P. Isaza, F. Linares and G. Ponce, On the propagation of regularity and decay of solutions to the $ k$-generalized Korteweg-de Vries equation, Comm. Partial Differential Equations, 2015, 40(7), 1336–1364. doi: 10.1080/03605302.2014.985794 CrossRef $ k$-generalized Korteweg-de Vries equation" target="_blank">Google Scholar
[4]	P. Isaza, F. Linares and G. Ponce, On the propagation of regularity of solutions of the Kadomtsev-Petviashvili equation, SIAM J. Math. Anal., 2016, 48(2), 1006–1024. doi: 10.1137/15M1012098 CrossRef Google Scholar
[5]	P. Isaza, F. Linares and G. Ponce, On the propagation of regularity in solutions of the Benjamin-Ono equation, J. Funct. Anal., 2016, 270(3), 976–1000. doi: 10.1016/j.jfa.2015.11.009 CrossRef Google Scholar
[6]	S. Kwon, On the fifth-order KdV equation: local well-posedness and lack of uniform continuity of the solution map, J. Differential Equations, 2008, 245(9), 2627–2659. doi: 10.1016/j.jde.2008.03.020 CrossRef Google Scholar
[7]	C. E. Kenig, F. Linares, G. Ponce and L. Vega, On the regularity of solutions to the $ k$-generalized Korteweg-de Vries equation, Proc. Amer. Math. Soc., 2018, 146, 3759–3766. doi: 10.1090/proc/13506 CrossRef $ k$-generalized Korteweg-de Vries equation" target="_blank">Google Scholar
[8]	F. Linares and G. Ponce, On special regularity properties of solutions of the Zakharov-Kuznetsov equation, Comm. Pure Appl. Math., 2018, 17(4), 1561–1572. Google Scholar
[9]	F. Linares, G. Ponce and D. Smith, On the regularity of solutions to a class of nonlinear dispersive equations, Math. Ann., 2017, 369, 797–837. doi: 10.1007/s00208-016-1452-8 CrossRef Google Scholar
[10]	A. J. Mendez, On the propagation of regularity in solutions of the dispersive generalized Benjamin-Ono equation, Anal. PDE, 2020, 13(8), 2399–2440. doi: 10.2140/apde.2020.13.2399 CrossRef Google Scholar
[11]	A. J. Mendez, On the propagation of regularity for solutions of the fractional Korteweg-de Vries equation, J. Differential Equations, 2020, 269(11), 9051–9089. doi: 10.1016/j.jde.2020.06.027 CrossRef Google Scholar
[12]	A. J. Mendez, On the propagation of regularity for solutions of the Zakharov-Kuznetsov equation, arXiv preprint, 2020, DOI: 10.48550/arXiv.2008.11252. Google Scholar
[13]	C. Mu$\tilde{n}$oz, G. Ponce and J. C. Saut, On the long time behavior of solutions to the Intermediate Long Wave equation, SIAM J. Math. Anal., 2021, 53(1), 1029–1048. doi: 10.1137/19M1293181 CrossRef Google Scholar
[14]	A. C. Nascimento, On the propagation of regularities in solutions of the fifth order Kadomtsev-Petviashvili II equation, J. Math. Anal. Appl., 2019, 478(1), 156–181. doi: 10.1016/j.jmaa.2019.05.024 CrossRef Google Scholar
[15]	A. C. Nascimento, On special regularity properties of solutions of the Benjamin-Ono-Zakharov-Kuznetsov (BO-ZK) equation, Comm. Pure Appl. Math., 2020, 19(9), 4285–4325. Google Scholar
[16]	J. I. Segata and D. L. Smith, Propagation of regularity and persistence of decay for fifth order dispersive models, J. Dynam. Differential Equations, 2017, 29, 701–736. doi: 10.1007/s10884-015-9499-x CrossRef Google Scholar