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A comparison principle for Hamilton-Jacobi equation with moving in time boundary
Asymptotic behavior of solutions to a higher-order KdV-type equation with critical nonlinearity
Division of Mathematics and Physics, Faculty of Engineering, Shinshu University, 4-17-1 Wakasato, Nagano City 380-8553, Japan |
$ \partial_t u + \frac{1}{ {\mathfrak{m}}} | \partial_x|^{ {\mathfrak{m}}-1} \partial_x u = \partial_x (u^{ {\mathfrak{m}}}) $ |
$ {\mathfrak{m}} \ge 4 $ |
References:
[1] |
P. Deift and X. Zhou,
A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation, Ann. of Math. (2), 137 (1993), 295-368.
doi: 10.2307/2946540. |
[2] |
B. Dodson, Global well-posedness and scattering for the defocusing, mass-critical generalized KdV equation, Ann. PDE, 3 (2017), Art. 5, 35 pp.
doi: 10.1007/s40818-017-0025-9. |
[3] |
P. Germain, F. Pusateri and F. Rousset,
Asymptotic stability of solitons for mKdV, Adv. Math., 299 (2016), 272-330.
doi: 10.1016/j.aim.2016.04.023. |
[4] |
A. Grünrock,
On the hierarchies of higher order mKdV and KdV equations, Cent. Eur. J. Math., 8 (2010), 500-536.
doi: 10.2478/s11533-010-0024-5. |
[5] |
B. Harrop-Griffiths,
Long time behavior of solutions to the mKdV, Comm. Partial Differential Equations, 41 (2016), 282-317.
doi: 10.1080/03605302.2015.1114495. |
[6] |
B. Harrop-Griffiths, M. Ifrim and D. Tataru,
The lifespan of small data solutions to the KP-I, Int. Math. Res. Not. IMRN, (2017), 1-28.
|
[7] |
N. Hayashi and P. I. Naumkin,
Large time asymptotics of solutions to the generalized Korteweg-de Vries equation, J. Funct. Anal., 159 (1998), 110-136.
doi: 10.1006/jfan.1998.3291. |
[8] |
N. Hayashi and P. I. Naumkin,
Large time behavior of solutions for the modified Korteweg-de Vries equation, Internat. Math. Res. Notices, (1999), 395-418.
doi: 10.1155/S1073792899000203. |
[9] |
N. Hayashi and P. I. Naumkin,
On the modified Korteweg-de Vries equation, Math. Phys. Anal. Geom., 4 (2001), 197-227.
doi: 10.1023/A:1012953917956. |
[10] |
N. Hayashi and P. I. Naumkin,
Large time asymptotics for the fourth-order nonlinear Schrödinger equation, J. Differential Equations, 258 (2015), 880-905.
doi: 10.1016/j.jde.2014.10.007. |
[11] |
N. Hayashi and P. I. Naumkin,
Factorization technique for the fourth-order nonlinear Schrödinger equation, Z. Angew. Math. Phys., 66 (2015), 2343-2377.
doi: 10.1007/s00033-015-0524-z. |
[12] |
N. Hayashi and P. I. Naumkin,
Global existence and asymptotic behavior of solutions to the fourth-order nonlinear Schrödinger equation in the critical case, Nonlinear Anal., 116 (2015), 112-131.
doi: 10.1016/j.na.2014.12.024. |
[13] |
N. Hayashi and P. I. Naumkin,
Factorization technique for the modified Korteweg e Vries equation, SUT J. Math., 52 (2016), 49-95.
|
[14] |
H. Hirayama and M. Okamoto,
Well-posedness and scattering for fourth order nonlinear Schrödinger type equations at the scaling critical regularity, Commun. Pure Appl. Anal., 15 (2016), 831-851.
doi: 10.3934/cpaa.2016.15.831. |
[15] |
M. Ifrim and D. Tataru,
Global bounds for the cubic nonlinear Schrödinger equation (NLS) in one space dimension, Nonlinearity, 28 (2015), 2661-2675.
doi: 10.1088/0951-7715/28/8/2661. |
[16] |
T. Kato and G. Ponce,
Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907.
doi: 10.1002/cpa.3160410704. |
[17] |
C. E. Kenig, G. Ponce and L. Vega,
On the (generalized) Korteweg-de Vries equation, Duke Math. J., 59 (1989), 585-610.
doi: 10.1215/S0012-7094-89-05927-9. |
[18] |
C. E. 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. |
[19] |
C. E. Kenig, G. Ponce and L. Vega,
Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46 (1993), 527-620.
doi: 10.1002/cpa.3160460405. |
[20] |
C. E. Kenig, G. Ponce and L. Vega, On the hierarchy of the generalized KdV equations, Singular Limits of Dispersive Waves (Lyon, 1991), 347–356, NATO Adv. Sci. Inst. Ser. B Phys., 320, Plenum, New York, 1994. |
[21] |
C. E. Kenig, G. Ponce, and L. Vega, On the concentration of blow up solutions for the generalized KdV equation critical in $L^2$, Nonlinear Wave Equations (Providence, RI, 1998), 131–156, Contemp. Math., 263, Amer. Math. Soc., Providence, RI, 2000.
doi: 10.1090/conm/263/04195. |
[22] |
H. Koch and J. Marzuola,
Small data scattering and soliton stability in $\dot{H}^{-1/6}$ for the quartic KdV equation, Anal. PDE, 5 (2012), 145-198.
doi: 10.2140/apde.2012.5.145. |
[23] |
F. Merle,
Existence of blow-up solutions in the energy space for the critical generalized KdV equation, J. Amer. Math. Soc., 14 (2001), 555-578.
doi: 10.1090/S0894-0347-01-00369-1. |
[24] |
M. Okamoto, Large time asymptotics of solutions to the short-pulse equation, NoDEA Nonlinear Differential Equations Appl., 24 (2017), Art. 42, 24 pp.
doi: 10.1007/s00030-017-0464-8. |
[25] |
M. Okamoto,
Long-time behavior of solutions to the fifth-order modified KdV-type equation, Adv. Differential Equations, 23 (2018), 751-792.
|
[26] |
A. Sidi, C. Sulem and P. L. Sulem,
On the long time behaviour of a generalized KdV equation, Acta Appl. Math., 7 (1986), 35-47.
doi: 10.1007/BF00046976. |
[27] |
E. M. Stein,
Interpolation of linear operators, Trans. Amer. Math. Soc., 83 (1956), 482-492.
doi: 10.1090/S0002-9947-1956-0082586-0. |
[28] |
T. Tao,
Scattering for the quartic generalised Korteweg-de Vries equation, J. Differential Equations, 232 (2007), 623-651.
doi: 10.1016/j.jde.2006.07.019. |
show all references
References:
[1] |
P. Deift and X. Zhou,
A steepest descent method for oscillatory Riemann-Hilbert problems. Asymptotics for the MKdV equation, Ann. of Math. (2), 137 (1993), 295-368.
doi: 10.2307/2946540. |
[2] |
B. Dodson, Global well-posedness and scattering for the defocusing, mass-critical generalized KdV equation, Ann. PDE, 3 (2017), Art. 5, 35 pp.
doi: 10.1007/s40818-017-0025-9. |
[3] |
P. Germain, F. Pusateri and F. Rousset,
Asymptotic stability of solitons for mKdV, Adv. Math., 299 (2016), 272-330.
doi: 10.1016/j.aim.2016.04.023. |
[4] |
A. Grünrock,
On the hierarchies of higher order mKdV and KdV equations, Cent. Eur. J. Math., 8 (2010), 500-536.
doi: 10.2478/s11533-010-0024-5. |
[5] |
B. Harrop-Griffiths,
Long time behavior of solutions to the mKdV, Comm. Partial Differential Equations, 41 (2016), 282-317.
doi: 10.1080/03605302.2015.1114495. |
[6] |
B. Harrop-Griffiths, M. Ifrim and D. Tataru,
The lifespan of small data solutions to the KP-I, Int. Math. Res. Not. IMRN, (2017), 1-28.
|
[7] |
N. Hayashi and P. I. Naumkin,
Large time asymptotics of solutions to the generalized Korteweg-de Vries equation, J. Funct. Anal., 159 (1998), 110-136.
doi: 10.1006/jfan.1998.3291. |
[8] |
N. Hayashi and P. I. Naumkin,
Large time behavior of solutions for the modified Korteweg-de Vries equation, Internat. Math. Res. Notices, (1999), 395-418.
doi: 10.1155/S1073792899000203. |
[9] |
N. Hayashi and P. I. Naumkin,
On the modified Korteweg-de Vries equation, Math. Phys. Anal. Geom., 4 (2001), 197-227.
doi: 10.1023/A:1012953917956. |
[10] |
N. Hayashi and P. I. Naumkin,
Large time asymptotics for the fourth-order nonlinear Schrödinger equation, J. Differential Equations, 258 (2015), 880-905.
doi: 10.1016/j.jde.2014.10.007. |
[11] |
N. Hayashi and P. I. Naumkin,
Factorization technique for the fourth-order nonlinear Schrödinger equation, Z. Angew. Math. Phys., 66 (2015), 2343-2377.
doi: 10.1007/s00033-015-0524-z. |
[12] |
N. Hayashi and P. I. Naumkin,
Global existence and asymptotic behavior of solutions to the fourth-order nonlinear Schrödinger equation in the critical case, Nonlinear Anal., 116 (2015), 112-131.
doi: 10.1016/j.na.2014.12.024. |
[13] |
N. Hayashi and P. I. Naumkin,
Factorization technique for the modified Korteweg e Vries equation, SUT J. Math., 52 (2016), 49-95.
|
[14] |
H. Hirayama and M. Okamoto,
Well-posedness and scattering for fourth order nonlinear Schrödinger type equations at the scaling critical regularity, Commun. Pure Appl. Anal., 15 (2016), 831-851.
doi: 10.3934/cpaa.2016.15.831. |
[15] |
M. Ifrim and D. Tataru,
Global bounds for the cubic nonlinear Schrödinger equation (NLS) in one space dimension, Nonlinearity, 28 (2015), 2661-2675.
doi: 10.1088/0951-7715/28/8/2661. |
[16] |
T. Kato and G. Ponce,
Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math., 41 (1988), 891-907.
doi: 10.1002/cpa.3160410704. |
[17] |
C. E. Kenig, G. Ponce and L. Vega,
On the (generalized) Korteweg-de Vries equation, Duke Math. J., 59 (1989), 585-610.
doi: 10.1215/S0012-7094-89-05927-9. |
[18] |
C. E. 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. |
[19] |
C. E. Kenig, G. Ponce and L. Vega,
Well-posedness and scattering results for the generalized Korteweg-de Vries equation via the contraction principle, Comm. Pure Appl. Math., 46 (1993), 527-620.
doi: 10.1002/cpa.3160460405. |
[20] |
C. E. Kenig, G. Ponce and L. Vega, On the hierarchy of the generalized KdV equations, Singular Limits of Dispersive Waves (Lyon, 1991), 347–356, NATO Adv. Sci. Inst. Ser. B Phys., 320, Plenum, New York, 1994. |
[21] |
C. E. Kenig, G. Ponce, and L. Vega, On the concentration of blow up solutions for the generalized KdV equation critical in $L^2$, Nonlinear Wave Equations (Providence, RI, 1998), 131–156, Contemp. Math., 263, Amer. Math. Soc., Providence, RI, 2000.
doi: 10.1090/conm/263/04195. |
[22] |
H. Koch and J. Marzuola,
Small data scattering and soliton stability in $\dot{H}^{-1/6}$ for the quartic KdV equation, Anal. PDE, 5 (2012), 145-198.
doi: 10.2140/apde.2012.5.145. |
[23] |
F. Merle,
Existence of blow-up solutions in the energy space for the critical generalized KdV equation, J. Amer. Math. Soc., 14 (2001), 555-578.
doi: 10.1090/S0894-0347-01-00369-1. |
[24] |
M. Okamoto, Large time asymptotics of solutions to the short-pulse equation, NoDEA Nonlinear Differential Equations Appl., 24 (2017), Art. 42, 24 pp.
doi: 10.1007/s00030-017-0464-8. |
[25] |
M. Okamoto,
Long-time behavior of solutions to the fifth-order modified KdV-type equation, Adv. Differential Equations, 23 (2018), 751-792.
|
[26] |
A. Sidi, C. Sulem and P. L. Sulem,
On the long time behaviour of a generalized KdV equation, Acta Appl. Math., 7 (1986), 35-47.
doi: 10.1007/BF00046976. |
[27] |
E. M. Stein,
Interpolation of linear operators, Trans. Amer. Math. Soc., 83 (1956), 482-492.
doi: 10.1090/S0002-9947-1956-0082586-0. |
[28] |
T. Tao,
Scattering for the quartic generalised Korteweg-de Vries equation, J. Differential Equations, 232 (2007), 623-651.
doi: 10.1016/j.jde.2006.07.019. |
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