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September  2019, 18(5): 2607-2661. doi: 10.3934/cpaa.2019117

## Local well-posedness of the fifth-order KdV-type equations on the half-line

 1 Instituto de Matemática, Universidade Federal de Alagoas, Av. Lourival Melo Mota, s/n Tabuleiro do Martins, Maceió, Alagoas, Brazil 2 Facultad de Matemáticas, Pontificia Universidad Católica de Chile, Campus San Joaquín, Avda. Vicuña Mackenna 4860, Santiago, Chile and Institute of Pure and Applied Mathematics, Chonbuk National University

* Corresponding author

Received  August 2018 Revised  January 2019 Published  April 2019

Fund Project: The second author is supported by FONDECYT de Postdoctorado 2017 Proyecto No. 3170067

This paper is a continuation of authors' previous work [6]. We extend the argument [6] to fifth-order KdV-type equations with different nonlinearities, in specific, where the scaling argument does not hold. We establish the $X^{s,b}$ nonlinear estimates for $b < \frac12$, which is almost optimal compared to the standard $X^{s,b}$ nonlinear estimates for $b > \frac12$ [8,17]. As an immediate conclusion, we prove the local well-posedness of the initial-boundary value problem (IBVP) for fifth-order KdV-type equations on the right half-line and the left half-line.

Citation: Márcio Cavalcante, Chulkwang Kwak. Local well-posedness of the fifth-order KdV-type equations on the half-line. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2607-2661. doi: 10.3934/cpaa.2019117
##### References:
 [1] L. Abramyan and Y. Stepanyants, The structure of two-dimensional solitons in media with anomalously small dispersion, Sov. Phys. JETP, 61 (1985), 963-966. Google Scholar [2] J. L. Bona, S. M. Sun and B.Y. Zhang, Boundary smoothing properties of the Korteweg-de Vries equation in a quarter plane and applications, Dyn. Partial Differ. Equ., 3 (2006), 1-69. doi: 10.4310/DPDE.2006.v3.n1.a1. Google Scholar [3] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. Parts Ⅰ, Ⅱ, Geom. Funct. Anal., 3 (1993), 107-262. doi: 10.1007/BF01896020. Google Scholar [4] M. Cavalcante, The initial-boundary-value problem for some quadratic nonlinear Schrödinger equations on the half-line, Differential and Integral Equations, 30 (2017), 521-554. Google Scholar [5] M. Cavalcante and A. J. Corcho, The initial boundary value problem for the Schrödinger-Korteweg-de Vries system on the half-line, Communications in Contemporary Mathematics, scheduled for publication (online ready). doi: 10.1142/S0219199718500669. Google Scholar [6] M. Cavalcante and C. Kwak, The initial-boundary value problem for the Kawahara equation on the half-line, preprint, arXiv: 1805.05229.Google Scholar [7] W. Chen and Z. Guo, Global well-posedness and Ⅰ-method for the fifth-order Korteweg-de Vries equation, J. Anal. Math., 114 (2011), 121-156. doi: 10.1007/s11854-011-0014-y. Google Scholar [8] W. Chen, Z. Guo and J. Liu, Sharp local well-posedness for a fifth-order shallow water wave equation, J. Math. Anal. Appl., 369 (2010), 133-143. doi: 10.1016/j.jmaa.2010.02.023. Google Scholar [9] W. Chen and J. Liu, Well-posedness and ill-posedness for a fifth-order shallow water wave equation, Nonlinear Anal., 72 (2010), 2412-2420. doi: 10.1016/j.na.2009.11.003. Google Scholar [10] W. Chen, J. Li, C. Miao and J. Wu, Low regularity solutions of two fifth-order KdV type equations, J. Anal. Math., 107 (2009), 221-238. doi: 10.1007/s11854-009-0009-0. Google Scholar [11] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $\mathbb R$ and ${\mathbb T}$, J. Amer. Math. Soc., 16 (2003), 705-749. doi: 10.1090/S0894-0347-03-00421-1. Google Scholar [12] J. Colliander and C. Kenig, The generalized Korteweg-de Vries equation on the half line, Comm. Partial Differential Equations, 27 (2002), 2187-2266. doi: 10.1081/PDE-120016157. Google Scholar [13] S. Cui and S. Tao, Strichartz estimates for dispersive equations and solvability of the Kawahara equation, J. Math. Anal. Appl., 304 (2005), 683-702. doi: 10.1016/j.jmaa.2004.09.049. Google Scholar [14] R. Grimshaw and N. Joshi, Weakly nonlocal waves in a singularly perturbed Korteweg-de Vries equation, SIAM J. Appl. Math., 55 (1995), 124-135. doi: 10.1137/S0036139993243825. Google Scholar [15] J. Holmer, The initial-boundary value problem for the Korteweg-de Vries equation, Communications in Partial Differential Equations, 31 (2006), 1151-1190. doi: 10.1080/03605300600718503. Google Scholar [16] J. K. Hunter and J. Scheurle, Existence of perturbed solitary wave solutions to a model equation for water waves, Physica D, 32 (1988), 253-268. doi: 10.1016/0167-2789(88)90054-1. Google Scholar [17] Y. Jia and Z. Huo, Well-posedness for the fifth-order shallow water equations, J. Differential Equations, 246 (2009), 2448-2467. doi: 10.1016/j.jde.2008.10.027. Google Scholar [18] D. Jerison and C. Kenig, The inhomogeneous Dirichlet Problem in Lipschitz Domains, J. Funct. Anal., 130 (1995), 161-219. doi: 10.1006/jfan.1995.1067. Google Scholar [19] T. Kawahara, Oscillatory solitary waves in dispersive media, Journal of Physical Society Japan, 33 (1972), 260-264. Google Scholar [20] C. Kenig and D. Pilod, Well-posedness for the fifth-order KdV equation in the energy space, Trans. Amer. Math. Soc., 367 (2015), 2551-2612. doi: 10.1090/S0002-9947-2014-05982-5. Google Scholar [21] C. Kenig, G. Ponce and L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J., 40 (1991), 33-69. doi: 10.1512/iumj.1991.40.40003. Google Scholar [22] C. Kwak, Low regularity Cauchy problem for the fifth-order modified KdV equations on $\mathbb{T}$, Journal of Hyperbolic Differential Equations, 15 (2018), 463-557. doi: 10.1142/S0219891618500170. Google Scholar [23] N. Larkin and G. Doronin, Kawahara equation in a quarter-plane and in a finite domain, Bol. Soc. Parana. Mat., 25 (2007), 9-16. doi: 10.5269/bspm.v25i1-2.7421. Google Scholar [24] N. Larkin and M. Simões, The Kawahara equation on bounded intervals and on a half-line, Nonlinear Analysis, 127 (2015), 397-412. doi: 10.1016/j.na.2015.07.008. Google Scholar [25] K. Sangare, A mixed problem in a half-strip for a generalized Kawahara equation in the space of infinitely differentiable exponentially decreasing functions, Vestnik RUDN Ser. Mat., 10 (2003), 91-107. Google Scholar [26] K. Sangare and A. Faminskii, Weak solutions of a mixed problem in a halfstrip for a generalized Kawahara equation, Matematicheskie Zametki, 85 (2009), 98-109. doi: 10.1134/S000143460901009X. Google Scholar [27] E. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, 43. Monographs in Harmonic Analysis, Ⅲ. Princeton University Press 1993. Google Scholar [28] E. Stein and R. Shakarchi, Complex Analysis, Princeton Lectures in Analysis, Ⅱ. Princeton University Press, 2003. Google Scholar [29] T. Tao, Multilinear weighted convolution of $L^2$ functions and applications to nonlinear dispersive equations, Amer. J. Math., 123 (2001), 839-908. Google Scholar [30] T. Tao, Nonlinear Dispersive Equations: Local and Global Analysis, CBMS Reg. Conf. Ser. Math., vol.106 (2006). doi: 10.1090/cbms/106. Google Scholar [31] S. Tao and S. Lu, An initial-boundary value problem for the modified Kawahara equation on the half line, Acta Math. Sinica (Chin. Ser.), 50 (2007), 241-254. Google Scholar [32] L. Tian, G. Gui and Y. Liu, On the Cauchy problem for the generalized shallow water wave equation, J. Differential Equations, 245 (2008), 1838-1852. doi: 10.1016/j.jde.2008.07.006. Google Scholar [33] W. Yan and Y. Li, Ill-posedness of modified Kawahara equation and Kaup-Kupershmidt equation, Acta Math. Sci. Ser. B (Engl. Ed.), 32 (2012), 710-716. doi: 10.1016/S0252-9602(12)60050-2. Google Scholar [34] W. Yan, Y. Li and X. Yang, The Cauchy problem for the modified Kawahara equation in Sobolev spaces with low regularity, Mathematical and Computer Modelling, 54 (2011), 1252-1261. doi: 10.1016/j.mcm.2011.03.036. Google Scholar

show all references

##### References:
 [1] L. Abramyan and Y. Stepanyants, The structure of two-dimensional solitons in media with anomalously small dispersion, Sov. Phys. JETP, 61 (1985), 963-966. Google Scholar [2] J. L. Bona, S. M. Sun and B.Y. Zhang, Boundary smoothing properties of the Korteweg-de Vries equation in a quarter plane and applications, Dyn. Partial Differ. Equ., 3 (2006), 1-69. doi: 10.4310/DPDE.2006.v3.n1.a1. Google Scholar [3] J. Bourgain, Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations. Parts Ⅰ, Ⅱ, Geom. Funct. Anal., 3 (1993), 107-262. doi: 10.1007/BF01896020. Google Scholar [4] M. Cavalcante, The initial-boundary-value problem for some quadratic nonlinear Schrödinger equations on the half-line, Differential and Integral Equations, 30 (2017), 521-554. Google Scholar [5] M. Cavalcante and A. J. Corcho, The initial boundary value problem for the Schrödinger-Korteweg-de Vries system on the half-line, Communications in Contemporary Mathematics, scheduled for publication (online ready). doi: 10.1142/S0219199718500669. Google Scholar [6] M. Cavalcante and C. Kwak, The initial-boundary value problem for the Kawahara equation on the half-line, preprint, arXiv: 1805.05229.Google Scholar [7] W. Chen and Z. Guo, Global well-posedness and Ⅰ-method for the fifth-order Korteweg-de Vries equation, J. Anal. Math., 114 (2011), 121-156. doi: 10.1007/s11854-011-0014-y. Google Scholar [8] W. Chen, Z. Guo and J. Liu, Sharp local well-posedness for a fifth-order shallow water wave equation, J. Math. Anal. Appl., 369 (2010), 133-143. doi: 10.1016/j.jmaa.2010.02.023. Google Scholar [9] W. Chen and J. Liu, Well-posedness and ill-posedness for a fifth-order shallow water wave equation, Nonlinear Anal., 72 (2010), 2412-2420. doi: 10.1016/j.na.2009.11.003. Google Scholar [10] W. Chen, J. Li, C. Miao and J. Wu, Low regularity solutions of two fifth-order KdV type equations, J. Anal. Math., 107 (2009), 221-238. doi: 10.1007/s11854-009-0009-0. Google Scholar [11] J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Sharp global well-posedness for KdV and modified KdV on $\mathbb R$ and ${\mathbb T}$, J. Amer. Math. Soc., 16 (2003), 705-749. doi: 10.1090/S0894-0347-03-00421-1. Google Scholar [12] J. Colliander and C. Kenig, The generalized Korteweg-de Vries equation on the half line, Comm. Partial Differential Equations, 27 (2002), 2187-2266. doi: 10.1081/PDE-120016157. Google Scholar [13] S. Cui and S. Tao, Strichartz estimates for dispersive equations and solvability of the Kawahara equation, J. Math. Anal. Appl., 304 (2005), 683-702. doi: 10.1016/j.jmaa.2004.09.049. Google Scholar [14] R. Grimshaw and N. Joshi, Weakly nonlocal waves in a singularly perturbed Korteweg-de Vries equation, SIAM J. Appl. Math., 55 (1995), 124-135. doi: 10.1137/S0036139993243825. Google Scholar [15] J. Holmer, The initial-boundary value problem for the Korteweg-de Vries equation, Communications in Partial Differential Equations, 31 (2006), 1151-1190. doi: 10.1080/03605300600718503. Google Scholar [16] J. K. Hunter and J. Scheurle, Existence of perturbed solitary wave solutions to a model equation for water waves, Physica D, 32 (1988), 253-268. doi: 10.1016/0167-2789(88)90054-1. Google Scholar [17] Y. Jia and Z. Huo, Well-posedness for the fifth-order shallow water equations, J. Differential Equations, 246 (2009), 2448-2467. doi: 10.1016/j.jde.2008.10.027. Google Scholar [18] D. Jerison and C. Kenig, The inhomogeneous Dirichlet Problem in Lipschitz Domains, J. Funct. Anal., 130 (1995), 161-219. doi: 10.1006/jfan.1995.1067. Google Scholar [19] T. Kawahara, Oscillatory solitary waves in dispersive media, Journal of Physical Society Japan, 33 (1972), 260-264. Google Scholar [20] C. Kenig and D. Pilod, Well-posedness for the fifth-order KdV equation in the energy space, Trans. Amer. Math. Soc., 367 (2015), 2551-2612. doi: 10.1090/S0002-9947-2014-05982-5. Google Scholar [21] C. Kenig, G. Ponce and L. Vega, Oscillatory integrals and regularity of dispersive equations, Indiana Univ. Math. J., 40 (1991), 33-69. doi: 10.1512/iumj.1991.40.40003. Google Scholar [22] C. Kwak, Low regularity Cauchy problem for the fifth-order modified KdV equations on $\mathbb{T}$, Journal of Hyperbolic Differential Equations, 15 (2018), 463-557. doi: 10.1142/S0219891618500170. Google Scholar [23] N. Larkin and G. Doronin, Kawahara equation in a quarter-plane and in a finite domain, Bol. Soc. Parana. Mat., 25 (2007), 9-16. doi: 10.5269/bspm.v25i1-2.7421. Google Scholar [24] N. Larkin and M. Simões, The Kawahara equation on bounded intervals and on a half-line, Nonlinear Analysis, 127 (2015), 397-412. doi: 10.1016/j.na.2015.07.008. Google Scholar [25] K. Sangare, A mixed problem in a half-strip for a generalized Kawahara equation in the space of infinitely differentiable exponentially decreasing functions, Vestnik RUDN Ser. Mat., 10 (2003), 91-107. Google Scholar [26] K. Sangare and A. Faminskii, Weak solutions of a mixed problem in a halfstrip for a generalized Kawahara equation, Matematicheskie Zametki, 85 (2009), 98-109. doi: 10.1134/S000143460901009X. Google Scholar [27] E. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series, 43. Monographs in Harmonic Analysis, Ⅲ. Princeton University Press 1993. Google Scholar [28] E. Stein and R. Shakarchi, Complex Analysis, Princeton Lectures in Analysis, Ⅱ. Princeton University Press, 2003. Google Scholar [29] T. Tao, Multilinear weighted convolution of $L^2$ functions and applications to nonlinear dispersive equations, Amer. J. Math., 123 (2001), 839-908. Google Scholar [30] T. Tao, Nonlinear Dispersive Equations: Local and Global Analysis, CBMS Reg. Conf. Ser. Math., vol.106 (2006). doi: 10.1090/cbms/106. Google Scholar [31] S. Tao and S. Lu, An initial-boundary value problem for the modified Kawahara equation on the half line, Acta Math. Sinica (Chin. Ser.), 50 (2007), 241-254. Google Scholar [32] L. Tian, G. Gui and Y. Liu, On the Cauchy problem for the generalized shallow water wave equation, J. Differential Equations, 245 (2008), 1838-1852. doi: 10.1016/j.jde.2008.07.006. Google Scholar [33] W. Yan and Y. Li, Ill-posedness of modified Kawahara equation and Kaup-Kupershmidt equation, Acta Math. Sci. Ser. B (Engl. Ed.), 32 (2012), 710-716. doi: 10.1016/S0252-9602(12)60050-2. Google Scholar [34] W. Yan, Y. Li and X. Yang, The Cauchy problem for the modified Kawahara equation in Sobolev spaces with low regularity, Mathematical and Computer Modelling, 54 (2011), 1252-1261. doi: 10.1016/j.mcm.2011.03.036. Google Scholar
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