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September  2019, 8(3): 567-601. doi: 10.3934/eect.2019027

Asymptotic behavior of solutions to a higher-order KdV-type equation with critical nonlinearity

Mamoru Okamoto ,

Division of Mathematics and Physics, Faculty of Engineering, Shinshu University, 4-17-1 Wakasato, Nagano City 380-8553, Japan

* Corresponding author: Mamoru Okamoto

Received  June 2018 Revised  November 2018 Published  May 2019

We consider the Cauchy problem of the higher-order KdV-type equation:
$ \partial_t u + \frac{1}{ {\mathfrak{m}}} | \partial_x|^{ {\mathfrak{m}}-1} \partial_x u = \partial_x (u^{ {\mathfrak{m}}}) $
where
$ {\mathfrak{m}} \ge 4 $
. The nonlinearity is critical in the sense of long-time behavior. Using the method of testing by wave packets, we prove that there exists a unique global solution of the Cauchy problem satisfying the same time decay estimate as that of linear solutions. Moreover, we divide the long-time behavior of the solution into three distinct regions.
Keywords: Higher-order KdV-type equation, asymptotic behavior, critical nonlinearity, self-similar solution.
Mathematics Subject Classification: Primary: 35Q53; Secondary: 35B40.
Citation: Mamoru Okamoto. Asymptotic behavior of solutions to a higher-order KdV-type equation with critical nonlinearity. Evolution Equations & Control Theory, 2019, 8 (3) : 567-601. doi: 10.3934/eect.2019027
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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[13]

N. Hayashi and P. I. Naumkin, Factorization technique for the modified Korteweg e Vries equation, SUT J. Math., 52 (2016), 49-95.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

M. Okamoto, Long-time behavior of solutions to the fifth-order modified KdV-type equation, Adv. Differential Equations, 23 (2018), 751-792.   Google Scholar

[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.  Google Scholar

[27]

E. M. Stein, Interpolation of linear operators, Trans. Amer. Math. Soc., 83 (1956), 482-492.  doi: 10.1090/S0002-9947-1956-0082586-0.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

N. Hayashi and P. I. Naumkin, Factorization technique for the modified Korteweg e Vries equation, SUT J. Math., 52 (2016), 49-95.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

M. Okamoto, Long-time behavior of solutions to the fifth-order modified KdV-type equation, Adv. Differential Equations, 23 (2018), 751-792.   Google Scholar

[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.  Google Scholar

[27]

E. M. Stein, Interpolation of linear operators, Trans. Amer. Math. Soc., 83 (1956), 482-492.  doi: 10.1090/S0002-9947-1956-0082586-0.  Google Scholar

[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.  Google Scholar

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