December  2016, 9(6): 2073-2094. doi: 10.3934/dcdss.2016085

Scattering theory for energy-supercritical Klein-Gordon equation

1. 

Institute of Applied Physics and Computational Mathematics, P. O. Box 8009, Beijing 100088, China

2. 

Université de Nice Sophia-Antipolis, 06108, Nice Cedex 02, France

Received  June 2015 Revised  August 2016 Published  November 2016

In this paper, we consider the question of the global well-posedness and scattering for the cubic Klein-Gordon equation $u_{t t}-\Delta u+u+|u|^2u=0$ in dimension $d\geq5$. We show that if the solution $u$ is apriorily bounded in the critical Sobolev space, that is, $(u, u_t)\in L_t^\infty(I; H^{s_c}_x(\mathbb{R}^d)\times H_x^{s_c-1}(\mathbb{R}^d))$ with $s_c:=\frac{d}2-1>1$, then $u$ is global and scatters. The impetus to consider this problem stems from a series of recent works for the energy-supercritical nonlinear wave equation and nonlinear Schrödinger equation. However, the scaling invariance is broken in the Klein-Gordon equation. We will utilize the concentration compactness ideas to show that the proof of the global well-posedness and scattering is reduced to disprove the existence of the scenario: soliton-like solutions. And such solutions are precluded by making use of the Morawetz inequality, finite speed of propagation and concentration of potential energy.
Citation: Changxing Miao, Jiqiang Zheng. Scattering theory for energy-supercritical Klein-Gordon equation. Discrete & Continuous Dynamical Systems - S, 2016, 9 (6) : 2073-2094. doi: 10.3934/dcdss.2016085
References:
[1]

J. Bergh and J. Löfström, Interpolation spaces, An introduciton. Grundlehren der Mathematischen Wissenschaften,, Springer-Verlag, (1976).   Google Scholar

[2]

M. Birman and S. Solomjak, On estimates on singular number of integral operators III,, Vest. LSU Math., 2 (1975), 9.   Google Scholar

[3]

P. Brenner, On space-time means and everywhere defined scattering operators for nonlinear Klein-Gordon equations,, Math. Z., 186 (1984), 383.  doi: 10.1007/BF01174891.  Google Scholar

[4]

P. Brenner, On scattering and everywhere defined scattering operators for nonlinear Klein-Gordon equtaons,, J. Differential Equations, 56 (1985), 310.  doi: 10.1016/0022-0396(85)90083-X.  Google Scholar

[5]

A. Bulut, Global well-posedness and scattering for the defocusing energy-supercritical cubic nonlinear wave equation,, J. Func. Anal., 263 (2012), 1609.  doi: 10.1016/j.jfa.2012.06.001.  Google Scholar

[6]

A. Bulut, The radial defocusing energy-supercritical cubic nonlinear wave equation in $\mathbbR^{1+5}$,, Nonlinearity, 27 (2014), 1859.  doi: 10.1088/0951-7715/27/8/1859.  Google Scholar

[7]

A. Bulut, The defocusing energy-supercritical cubic nonlinear wave equation in dimension five,, Tran. Amer. Math. Soc., 367 (2015), 6017.  doi: 10.1090/tran/6068.  Google Scholar

[8]

M. Christ and M. Weinstein, Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation,, J. Funct. Anal., 100 (1991), 87.  doi: 10.1016/0022-1236(91)90103-C.  Google Scholar

[9]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness and scattering for the energy-cirtical nonlinear Schrödinger equation in $\mathbbR^3$,, Annals of Math., 167 (2008), 767.  doi: 10.4007/annals.2008.167.767.  Google Scholar

[10]

B. Dodson, C. Miao, J. Murphy and J. Zheng, The defocusing quintic NLS in four space dimensions,, To appear in Annales de l'Institut Henri Pincare/Analyse non lineaire, ().  doi: 10.1016/j.anihpc.2016.05.004.  Google Scholar

[11]

T. Duyckaerts, C. Kenig and F. Merle, Scattering for radial, bounded solutions of focusing supercritical wave equations,, Int. Math. Res. Notices, 2014 (2014), 224.   Google Scholar

[12]

T. Duyckaerts, C. Kenig and F. Merle, Universality of blow-up profile for small radial type II blow-up solutions of the energy-critical wave equation,, J. Eur. Math. Soc., 13 (2011), 533.  doi: 10.4171/JEMS/261.  Google Scholar

[13]

J. Ginibre and G. Velo, Time decay of finite energy solutions of the nonlinear Klein-Gordon and Schrödinger equations,, Ann. Inst. H. Poincaré Phys. Théor., 43 (1985), 399.   Google Scholar

[14]

J. Ginibre and G. Velo, Generalized Strichartz inequalities for the wave equation,, J. Funct. Anal., 133 (1995), 50.  doi: 10.1006/jfan.1995.1119.  Google Scholar

[15]

S. Ibrahim, N. Masmoudi and K. Nakanishi, Scattering threshold for the focusing nonlinear Klein-Gordon equation,, Analysis and PDE., 4 (2011), 405.  doi: 10.2140/apde.2011.4.405.  Google Scholar

[16]

S. Ibrahim, N. Masmoudi and K. Nakanishi, Threshold solutions in the case of mass-shift for the critical Klein-Gordon equation,, Tran. Amer. Math. Soc., 366 (2014), 5653.  doi: 10.1090/S0002-9947-2014-05852-2.  Google Scholar

[17]

M. Keel and T. Tao, Endpoint Strichartz estimates,, Amer. J. Math., 120 (1998), 955.  doi: 10.1353/ajm.1998.0039.  Google Scholar

[18]

C. Kenig and F. Merle, Global well-posedness, scattering, and blow-up for the energy-critical focusing nonlinear Schrödinger equation in the radial case,, Invent. Math., 166 (2006), 645.  doi: 10.1007/s00222-006-0011-4.  Google Scholar

[19]

C. Kenig and F. Merle, Nondispersive radial solutions to energy supercritical nonlinaer wave equations, with applications,, Amer. J. Math., 133 (2011), 1029.  doi: 10.1353/ajm.2011.0029.  Google Scholar

[20]

C. Kenig and F. Merle, Radial solutions to energy supercritical wave equations in odd dimensions,, Disc. Cont. Dyn. Sys. A, 31 (2011), 1365.  doi: 10.3934/dcds.2011.31.1365.  Google Scholar

[21]

R. Killip and M. Visan, Energy-supercritical NLS: critical $\dotH^s$-bounds imply scattering,, Comm. Partial Differential Equations, 35 (2010), 945.  doi: 10.1080/03605301003717084.  Google Scholar

[22]

R. Killip and M. Visan, The defocusing energy-supercritical nonlinear wave equation in three space dimensions,, Trans. Amer. Math. Soc., 363 (2011), 3893.  doi: 10.1090/S0002-9947-2011-05400-0.  Google Scholar

[23]

R. Killip and M. Visan, The radial defocusing energy-supercritical nonlinear wave equation in all space dimensions,, Proc. Amer. Math. Soc., 139 (2011), 1805.  doi: 10.1090/S0002-9939-2010-10615-9.  Google Scholar

[24]

C. Miao, Modern Methods to the Nonlinear Wave Equations,, Monographs on Modern pure mathematics(Second Edition), (2010).   Google Scholar

[25]

C. Miao, J. Murphy and J. Zheng, The defocusing energy-supercritical NLS in four space dimensions,, J. Funct. Anal., 267 (2014), 1662.  doi: 10.1016/j.jfa.2014.06.016.  Google Scholar

[26]

C. Miao, B. Zhang and D. Fang, Global well-posedness for the Klein-Gordon equations below the energy norm,, Journal of Partial Differential Equations, 17 (2004), 97.   Google Scholar

[27]

C. Morawetz and W. A. Strauss, Decay and scattering of solutions of a nonlinear relativistic wave equation,, Comm. Pure Appl. Math., 25 (1972), 1.  doi: 10.1002/cpa.3160250103.  Google Scholar

[28]

K. Nakanishi, Scattering theory for nonlinear Klein-Gordon equation with Sobolev critical power,, Int. Math. Res. Notices, 1 (1999), 31.  doi: 10.1155/S1073792899000021.  Google Scholar

[29]

K. Nakanishi, Remarks on the energy scattering for nonlinear Klein-Gordon and Schröinger equations,, Tohoku Math. J., 53 (2001), 285.  doi: 10.2748/tmj/1178207482.  Google Scholar

[30]

M. Taylor, Tools for PDE,, Mathematical Surveys and Monographs, (2000).   Google Scholar

show all references

References:
[1]

J. Bergh and J. Löfström, Interpolation spaces, An introduciton. Grundlehren der Mathematischen Wissenschaften,, Springer-Verlag, (1976).   Google Scholar

[2]

M. Birman and S. Solomjak, On estimates on singular number of integral operators III,, Vest. LSU Math., 2 (1975), 9.   Google Scholar

[3]

P. Brenner, On space-time means and everywhere defined scattering operators for nonlinear Klein-Gordon equations,, Math. Z., 186 (1984), 383.  doi: 10.1007/BF01174891.  Google Scholar

[4]

P. Brenner, On scattering and everywhere defined scattering operators for nonlinear Klein-Gordon equtaons,, J. Differential Equations, 56 (1985), 310.  doi: 10.1016/0022-0396(85)90083-X.  Google Scholar

[5]

A. Bulut, Global well-posedness and scattering for the defocusing energy-supercritical cubic nonlinear wave equation,, J. Func. Anal., 263 (2012), 1609.  doi: 10.1016/j.jfa.2012.06.001.  Google Scholar

[6]

A. Bulut, The radial defocusing energy-supercritical cubic nonlinear wave equation in $\mathbbR^{1+5}$,, Nonlinearity, 27 (2014), 1859.  doi: 10.1088/0951-7715/27/8/1859.  Google Scholar

[7]

A. Bulut, The defocusing energy-supercritical cubic nonlinear wave equation in dimension five,, Tran. Amer. Math. Soc., 367 (2015), 6017.  doi: 10.1090/tran/6068.  Google Scholar

[8]

M. Christ and M. Weinstein, Dispersion of small amplitude solutions of the generalized Korteweg-de Vries equation,, J. Funct. Anal., 100 (1991), 87.  doi: 10.1016/0022-1236(91)90103-C.  Google Scholar

[9]

J. Colliander, M. Keel, G. Staffilani, H. Takaoka and T. Tao, Global well-posedness and scattering for the energy-cirtical nonlinear Schrödinger equation in $\mathbbR^3$,, Annals of Math., 167 (2008), 767.  doi: 10.4007/annals.2008.167.767.  Google Scholar

[10]

B. Dodson, C. Miao, J. Murphy and J. Zheng, The defocusing quintic NLS in four space dimensions,, To appear in Annales de l'Institut Henri Pincare/Analyse non lineaire, ().  doi: 10.1016/j.anihpc.2016.05.004.  Google Scholar

[11]

T. Duyckaerts, C. Kenig and F. Merle, Scattering for radial, bounded solutions of focusing supercritical wave equations,, Int. Math. Res. Notices, 2014 (2014), 224.   Google Scholar

[12]

T. Duyckaerts, C. Kenig and F. Merle, Universality of blow-up profile for small radial type II blow-up solutions of the energy-critical wave equation,, J. Eur. Math. Soc., 13 (2011), 533.  doi: 10.4171/JEMS/261.  Google Scholar

[13]

J. Ginibre and G. Velo, Time decay of finite energy solutions of the nonlinear Klein-Gordon and Schrödinger equations,, Ann. Inst. H. Poincaré Phys. Théor., 43 (1985), 399.   Google Scholar

[14]

J. Ginibre and G. Velo, Generalized Strichartz inequalities for the wave equation,, J. Funct. Anal., 133 (1995), 50.  doi: 10.1006/jfan.1995.1119.  Google Scholar

[15]

S. Ibrahim, N. Masmoudi and K. Nakanishi, Scattering threshold for the focusing nonlinear Klein-Gordon equation,, Analysis and PDE., 4 (2011), 405.  doi: 10.2140/apde.2011.4.405.  Google Scholar

[16]

S. Ibrahim, N. Masmoudi and K. Nakanishi, Threshold solutions in the case of mass-shift for the critical Klein-Gordon equation,, Tran. Amer. Math. Soc., 366 (2014), 5653.  doi: 10.1090/S0002-9947-2014-05852-2.  Google Scholar

[17]

M. Keel and T. Tao, Endpoint Strichartz estimates,, Amer. J. Math., 120 (1998), 955.  doi: 10.1353/ajm.1998.0039.  Google Scholar

[18]

C. Kenig and F. Merle, Global well-posedness, scattering, and blow-up for the energy-critical focusing nonlinear Schrödinger equation in the radial case,, Invent. Math., 166 (2006), 645.  doi: 10.1007/s00222-006-0011-4.  Google Scholar

[19]

C. Kenig and F. Merle, Nondispersive radial solutions to energy supercritical nonlinaer wave equations, with applications,, Amer. J. Math., 133 (2011), 1029.  doi: 10.1353/ajm.2011.0029.  Google Scholar

[20]

C. Kenig and F. Merle, Radial solutions to energy supercritical wave equations in odd dimensions,, Disc. Cont. Dyn. Sys. A, 31 (2011), 1365.  doi: 10.3934/dcds.2011.31.1365.  Google Scholar

[21]

R. Killip and M. Visan, Energy-supercritical NLS: critical $\dotH^s$-bounds imply scattering,, Comm. Partial Differential Equations, 35 (2010), 945.  doi: 10.1080/03605301003717084.  Google Scholar

[22]

R. Killip and M. Visan, The defocusing energy-supercritical nonlinear wave equation in three space dimensions,, Trans. Amer. Math. Soc., 363 (2011), 3893.  doi: 10.1090/S0002-9947-2011-05400-0.  Google Scholar

[23]

R. Killip and M. Visan, The radial defocusing energy-supercritical nonlinear wave equation in all space dimensions,, Proc. Amer. Math. Soc., 139 (2011), 1805.  doi: 10.1090/S0002-9939-2010-10615-9.  Google Scholar

[24]

C. Miao, Modern Methods to the Nonlinear Wave Equations,, Monographs on Modern pure mathematics(Second Edition), (2010).   Google Scholar

[25]

C. Miao, J. Murphy and J. Zheng, The defocusing energy-supercritical NLS in four space dimensions,, J. Funct. Anal., 267 (2014), 1662.  doi: 10.1016/j.jfa.2014.06.016.  Google Scholar

[26]

C. Miao, B. Zhang and D. Fang, Global well-posedness for the Klein-Gordon equations below the energy norm,, Journal of Partial Differential Equations, 17 (2004), 97.   Google Scholar

[27]

C. Morawetz and W. A. Strauss, Decay and scattering of solutions of a nonlinear relativistic wave equation,, Comm. Pure Appl. Math., 25 (1972), 1.  doi: 10.1002/cpa.3160250103.  Google Scholar

[28]

K. Nakanishi, Scattering theory for nonlinear Klein-Gordon equation with Sobolev critical power,, Int. Math. Res. Notices, 1 (1999), 31.  doi: 10.1155/S1073792899000021.  Google Scholar

[29]

K. Nakanishi, Remarks on the energy scattering for nonlinear Klein-Gordon and Schröinger equations,, Tohoku Math. J., 53 (2001), 285.  doi: 10.2748/tmj/1178207482.  Google Scholar

[30]

M. Taylor, Tools for PDE,, Mathematical Surveys and Monographs, (2000).   Google Scholar

[1]

Hironobu Sasaki. Remark on the scattering problem for the Klein-Gordon equation with power nonlinearity. Conference Publications, 2007, 2007 (Special) : 903-911. doi: 10.3934/proc.2007.2007.903

[2]

Satoshi Masaki, Jun-ichi Segata. Modified scattering for the Klein-Gordon equation with the critical nonlinearity in three dimensions. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1595-1611. doi: 10.3934/cpaa.2018076

[3]

Hironobu Sasaki. Small data scattering for the Klein-Gordon equation with cubic convolution nonlinearity. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 973-981. doi: 10.3934/dcds.2006.15.973

[4]

Hyeongjin Lee, Ihyeok Seo, Jihyeon Seok. Local smoothing and Strichartz estimates for the Klein-Gordon equation with the inverse-square potential. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 597-608. doi: 10.3934/dcds.2020024

[5]

Karen Yagdjian. The semilinear Klein-Gordon equation in de Sitter spacetime. Discrete & Continuous Dynamical Systems - S, 2009, 2 (3) : 679-696. doi: 10.3934/dcdss.2009.2.679

[6]

Aslihan Demirkaya, Panayotis G. Kevrekidis, Milena Stanislavova, Atanas Stefanov. Spectral stability analysis for standing waves of a perturbed Klein-Gordon equation. Conference Publications, 2015, 2015 (special) : 359-368. doi: 10.3934/proc.2015.0359

[7]

Chi-Kun Lin, Kung-Chien Wu. On the fluid dynamical approximation to the nonlinear Klein-Gordon equation. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2233-2251. doi: 10.3934/dcds.2012.32.2233

[8]

Jun Yang. Vortex structures for Klein-Gordon equation with Ginzburg-Landau nonlinearity. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2359-2388. doi: 10.3934/dcds.2014.34.2359

[9]

Elena Kopylova. On dispersion decay for 3D Klein-Gordon equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5765-5780. doi: 10.3934/dcds.2018251

[10]

Stefano Pasquali. A Nekhoroshev type theorem for the nonlinear Klein-Gordon equation with potential. Discrete & Continuous Dynamical Systems - B, 2018, 23 (9) : 3573-3594. doi: 10.3934/dcdsb.2017215

[11]

Baoxiang Wang. Scattering of solutions for critical and subcritical nonlinear Klein-Gordon equations in $H^s$. Discrete & Continuous Dynamical Systems - A, 1999, 5 (4) : 753-763. doi: 10.3934/dcds.1999.5.753

[12]

Xinliang An, Avy Soffer. Fermi's golden rule and $ H^1 $ scattering for nonlinear Klein-Gordon equations with metastable states. Discrete & Continuous Dynamical Systems - A, 2020, 40 (1) : 331-373. doi: 10.3934/dcds.2020013

[13]

Peter Bates, Chunlei Zhang. Traveling pulses for the Klein-Gordon equation on a lattice or continuum with long-range interaction. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 235-252. doi: 10.3934/dcds.2006.16.235

[14]

Neal Bez, Chris Jeavons. A sharp Sobolev-Strichartz estimate for the wave equation. Electronic Research Announcements, 2015, 22: 46-54. doi: 10.3934/era.2015.22.46

[15]

Benoît Grébert, Tiphaine Jézéquel, Laurent Thomann. Dynamics of Klein-Gordon on a compact surface near a homoclinic orbit. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3485-3510. doi: 10.3934/dcds.2014.34.3485

[16]

Masahito Ohta, Grozdena Todorova. Strong instability of standing waves for nonlinear Klein-Gordon equations. Discrete & Continuous Dynamical Systems - A, 2005, 12 (2) : 315-322. doi: 10.3934/dcds.2005.12.315

[17]

Marco Ghimenti, Stefan Le Coz, Marco Squassina. On the stability of standing waves of Klein-Gordon equations in a semiclassical regime. Discrete & Continuous Dynamical Systems - A, 2013, 33 (6) : 2389-2401. doi: 10.3934/dcds.2013.33.2389

[18]

Michinori Ishiwata, Makoto Nakamura, Hidemitsu Wadade. Remarks on the Cauchy problem of Klein-Gordon equations with weighted nonlinear terms. Discrete & Continuous Dynamical Systems - A, 2015, 35 (10) : 4889-4903. doi: 10.3934/dcds.2015.35.4889

[19]

M. Keel, Tristan Roy, Terence Tao. Global well-posedness of the Maxwell-Klein-Gordon equation below the energy norm. Discrete & Continuous Dynamical Systems - A, 2011, 30 (3) : 573-621. doi: 10.3934/dcds.2011.30.573

[20]

Soichiro Katayama. Global existence for systems of nonlinear wave and klein-gordon equations with compactly supported initial data. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1479-1497. doi: 10.3934/cpaa.2018071

2018 Impact Factor: 0.545

Metrics

  • PDF downloads (21)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]