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]

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[2]

M. Birman and S. Solomjak, On estimates on singular number of integral operators III, Vest. LSU Math., 2 (1975), 9-27. Google Scholar

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P. Brenner, On space-time means and everywhere defined scattering operators for nonlinear Klein-Gordon equations, Math. Z., 186 (1984), 383-391. doi: 10.1007/BF01174891.  Google Scholar

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A. Bulut, Global well-posedness and scattering for the defocusing energy-supercritical cubic nonlinear wave equation, J. Func. Anal., 263 (2012), 1609-1660. doi: 10.1016/j.jfa.2012.06.001.  Google Scholar

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A. Bulut, The defocusing energy-supercritical cubic nonlinear wave equation in dimension five, Tran. Amer. Math. Soc., 367 (2015), 6017-6061. doi: 10.1090/tran/6068.  Google Scholar

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T. Duyckaerts, C. Kenig and F. Merle, Scattering for radial, bounded solutions of focusing supercritical wave equations, Int. Math. Res. Notices, 2014 (2014), 224-258.  Google Scholar

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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), 533-599. doi: 10.4171/JEMS/261.  Google Scholar

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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), 399-442.  Google Scholar

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J. Ginibre and G. Velo, Generalized Strichartz inequalities for the wave equation, J. Funct. Anal., 133 (1995), 50-68. doi: 10.1006/jfan.1995.1119.  Google Scholar

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S. Ibrahim, N. Masmoudi and K. Nakanishi, Scattering threshold for the focusing nonlinear Klein-Gordon equation, Analysis and PDE., 4 (2011), 405-460. doi: 10.2140/apde.2011.4.405.  Google Scholar

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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), 5653-5669. doi: 10.1090/S0002-9947-2014-05852-2.  Google Scholar

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M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980. doi: 10.1353/ajm.1998.0039.  Google Scholar

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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), 645-675. doi: 10.1007/s00222-006-0011-4.  Google Scholar

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C. Kenig and F. Merle, Nondispersive radial solutions to energy supercritical nonlinaer wave equations, with applications, Amer. J. Math., 133 (2011), 1029-1065. doi: 10.1353/ajm.2011.0029.  Google Scholar

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C. Kenig and F. Merle, Radial solutions to energy supercritical wave equations in odd dimensions, Disc. Cont. Dyn. Sys. A, 31 (2011), 1365-1381. doi: 10.3934/dcds.2011.31.1365.  Google Scholar

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R. Killip and M. Visan, Energy-supercritical NLS: critical $\dotH^s$-bounds imply scattering, Comm. Partial Differential Equations, 35 (2010), 945-987. 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), 3893-3934. 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), 1805-1817. 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), No.133. Science Press, Beijing, 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), 1662-1724. 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), 97-121.  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), 1-31. 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), 31-60. 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), 285-303. doi: 10.2748/tmj/1178207482.  Google Scholar

[30]

M. Taylor, Tools for PDE, Mathematical Surveys and Monographs, vol. 81, Pseudodifferential operators, paradifferential operators, and layer potentials, American Mathematical Society, Providence, RI, 2000.  Google Scholar

show all references

References:
[1]

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

[2]

M. Birman and S. Solomjak, On estimates on singular number of integral operators III, Vest. LSU Math., 2 (1975), 9-27. Google Scholar

[3]

P. Brenner, On space-time means and everywhere defined scattering operators for nonlinear Klein-Gordon equations, Math. Z., 186 (1984), 383-391. 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), 310-344. 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), 1609-1660. 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), 1859-1877. 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), 6017-6061. 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), 87-109. 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), 767-865. 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), 224-258.  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), 533-599. 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), 399-442.  Google Scholar

[14]

J. Ginibre and G. Velo, Generalized Strichartz inequalities for the wave equation, J. Funct. Anal., 133 (1995), 50-68. 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), 405-460. 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), 5653-5669. doi: 10.1090/S0002-9947-2014-05852-2.  Google Scholar

[17]

M. Keel and T. Tao, Endpoint Strichartz estimates, Amer. J. Math., 120 (1998), 955-980. 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), 645-675. 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), 1029-1065. 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), 1365-1381. 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), 945-987. 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), 3893-3934. 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), 1805-1817. 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), No.133. Science Press, Beijing, 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), 1662-1724. 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), 97-121.  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), 1-31. 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), 31-60. 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), 285-303. doi: 10.2748/tmj/1178207482.  Google Scholar

[30]

M. Taylor, Tools for PDE, Mathematical Surveys and Monographs, vol. 81, Pseudodifferential operators, paradifferential operators, and layer potentials, American Mathematical Society, Providence, RI, 2000.  Google Scholar

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