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On extension problem, trace Hardy and Hardy's inequalities for some fractional Laplacians
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 |
This paper is a continuation of authors' previous work [
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.
|
[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. |
[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. |
[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.
|
[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. |
[6] |
M. Cavalcante and C. Kwak, The initial-boundary value problem for the Kawahara equation on the half-line, preprint, arXiv: 1805.05229. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[19] |
T. Kawahara,
Oscillatory solitary waves in dispersive media, Journal of Physical Society Japan, 33 (1972), 260-264.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[28] |
E. Stein and R. Shakarchi, Complex Analysis, Princeton Lectures in Analysis, Ⅱ. Princeton University Press, 2003. |
[29] |
T. Tao,
Multilinear weighted convolution of $L^2$ functions and applications to nonlinear dispersive equations, Amer. J. Math., 123 (2001), 839-908.
|
[30] |
T. Tao, Nonlinear Dispersive Equations: Local and Global Analysis, CBMS Reg. Conf. Ser. Math., vol.106 (2006).
doi: 10.1090/cbms/106. |
[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.
|
[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. |
[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. |
[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. |
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.
|
[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. |
[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. |
[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.
|
[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. |
[6] |
M. Cavalcante and C. Kwak, The initial-boundary value problem for the Kawahara equation on the half-line, preprint, arXiv: 1805.05229. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[19] |
T. Kawahara,
Oscillatory solitary waves in dispersive media, Journal of Physical Society Japan, 33 (1972), 260-264.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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.
|
[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. |
[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. |
[28] |
E. Stein and R. Shakarchi, Complex Analysis, Princeton Lectures in Analysis, Ⅱ. Princeton University Press, 2003. |
[29] |
T. Tao,
Multilinear weighted convolution of $L^2$ functions and applications to nonlinear dispersive equations, Amer. J. Math., 123 (2001), 839-908.
|
[30] |
T. Tao, Nonlinear Dispersive Equations: Local and Global Analysis, CBMS Reg. Conf. Ser. Math., vol.106 (2006).
doi: 10.1090/cbms/106. |
[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.
|
[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. |
[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. |
[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. |
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