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

Abstract

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 [Formula: see text]. We prove that any energy solution has a global bound of the [Formula: see text] space-time norm, and hence, scatters in [Formula: see text] as [Formula: see text]. 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 [Formula: see text] for NLKG with the exponential nonlinearity holds only for small initial data, we use the [Formula: see text]-norm for some [Formula: see text] instead of the [Formula: see text]-norm, where [Formula: see text] denotes the [Formula: see text]th order [Formula: see text]-based Sobolev space.