# American Institute of Mathematical Sciences

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, 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

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
 [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, 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, 2020, 40 (1) : 597-608. doi: 10.3934/dcds.2020024 [5] Milena Dimova, Natalia Kolkovska, Nikolai Kutev. Global behavior of the solutions to nonlinear Klein-Gordon equation with critical initial energy. Electronic Research Archive, 2020, 28 (2) : 671-689. doi: 10.3934/era.2020035 [6] 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 [7] 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 [8] 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 [9] Elena Kopylova. On dispersion decay for 3D Klein-Gordon equation. Discrete & Continuous Dynamical Systems, 2018, 38 (11) : 5765-5780. doi: 10.3934/dcds.2018251 [10] Chi-Kun Lin, Kung-Chien Wu. On the fluid dynamical approximation to the nonlinear Klein-Gordon equation. Discrete & Continuous Dynamical Systems, 2012, 32 (6) : 2233-2251. doi: 10.3934/dcds.2012.32.2233 [11] Jun Yang. Vortex structures for Klein-Gordon equation with Ginzburg-Landau nonlinearity. Discrete & Continuous Dynamical Systems, 2014, 34 (5) : 2359-2388. doi: 10.3934/dcds.2014.34.2359 [12] Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (5) : 1649-1672. doi: 10.3934/dcdss.2020448 [13] Jungkwon Kim, Hyeongjin Lee, Ihyeok Seo, Jihyeon Seok. On Morawetz estimates with time-dependent weights for the Klein-Gordon equation. Discrete & Continuous Dynamical Systems, 2020, 40 (11) : 6275-6288. doi: 10.3934/dcds.2020279 [14] Qinghua Luo. Damped Klein-Gordon equation with variable diffusion coefficient. Communications on Pure & Applied Analysis, 2021, 20 (11) : 3959-3974. doi: 10.3934/cpaa.2021139 [15] Baoxiang Wang. Scattering of solutions for critical and subcritical nonlinear Klein-Gordon equations in $H^s$. Discrete & Continuous Dynamical Systems, 1999, 5 (4) : 753-763. doi: 10.3934/dcds.1999.5.753 [16] Marcelo M. Cavalcanti, Leonel G. Delatorre, Daiane C. Soares, Victor Hugo Gonzalez Martinez, Janaina P. Zanchetta. Uniform stabilization of the Klein-Gordon system. Communications on Pure & Applied Analysis, 2020, 19 (11) : 5131-5156. doi: 10.3934/cpaa.2020230 [17] Xinliang An, Avy Soffer. Fermi's golden rule and $H^1$ scattering for nonlinear Klein-Gordon equations with metastable states. Discrete & Continuous Dynamical Systems, 2020, 40 (1) : 331-373. doi: 10.3934/dcds.2020013 [18] Katharina Schratz, Xiaofei Zhao. On comparison of asymptotic expansion techniques for nonlinear Klein-Gordon equation in the nonrelativistic limit regime. Discrete & Continuous Dynamical Systems - B, 2020, 25 (8) : 2841-2865. doi: 10.3934/dcdsb.2020043 [19] Zheng Han, Daoyuan Fang. Almost global existence for the Klein-Gordon equation with the Kirchhoff-type nonlinearity. Communications on Pure & Applied Analysis, 2021, 20 (2) : 737-754. doi: 10.3934/cpaa.2020287 [20] Peter Bates, Chunlei Zhang. Traveling pulses for the Klein-Gordon equation on a lattice or continuum with long-range interaction. Discrete & Continuous Dynamical Systems, 2006, 16 (1) : 235-252. doi: 10.3934/dcds.2006.16.235

2020 Impact Factor: 2.425