Scattering for the one-dimensional Klein-Gordon equation with exponential nonlinearity

被引:2
|
作者
Ikeda, Masahiro [1 ,2 ]
Inui, Takahisa [3 ]
Okamoto, Mamoru [4 ]
机构
[1] Keio Univ, Fac Sci & Technol, Dept Math, Kohoku Ku, 3-14-1 Hiyoshi, Yokohama, Kanagawa 2238522, Japan
[2] RIKEN, Ctr Adv Intelligence Project, Wako, Saitama, Japan
[3] Osaka Univ, Grad Sch Sci, Dept Math, Toyonaka, Osaka 5600043, Japan
[4] Shinshu Univ, Fac Engn, Div Math & Phys, 4-17-1 Wakasato, Nagano 3808553, Japan
关键词
Klein-Gordon equation; exponential nonlinearity; scattering; GLOBAL WELL-POSEDNESS; SCHRODINGER-EQUATIONS; REGULARITY; SPACE; MASS;
D O I
10.1142/S0219891620500083
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
We consider the asymptotic behavior of solutions to the Cauchy problem for the defocusing nonlinear Klein-Gordon equation (NLKG) with exponential nonlinearity in the one spatial dimension with data in the energy space H-1 (R) x L-2 (R). We prove that any energy solution has a global bound of the L-t,x(6) space-time norm, and hence, scatters in H-1 (R) x L-2 (R) as t -> +/-infinity. The proof is based on the argument by Killip-Stovall-Visan (Trans. Amer. Math. Soc. 364(3) (2012) 1571-1631). However, since well-posedness in H-1/2 (R) x H-1/2 (R) for NLKG with the exponential nonlinearity holds only for small initial data, we use the (LtWx8-1/2,6)-W-6-norm for some s > 1/2 instead of the L-t,x(6)-norm, where W-x(8,p) denotes the sth order L-p-based Sobolev space.
引用
收藏
页码:295 / 354
页数:60
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