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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. BonaS. 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. ChenZ. 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. ChenJ. LiC. 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. CollianderM. KeelG. StaffilaniH. 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

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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. KenigG. 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. TianG. 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. YanY. 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. BonaS. 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. ChenZ. 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. ChenJ. LiC. 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. CollianderM. KeelG. StaffilaniH. 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. KenigG. 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. TianG. 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. YanY. 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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