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

## TL;DR

This paper proves that solutions to the one-dimensional defocusing nonlinear Klein-Gordon equation with exponential nonlinearity scatter in the energy space, using a modified norm approach due to well-posedness constraints.

## Contribution

It extends scattering results to exponential nonlinearities in 1D Klein-Gordon equations by adapting the argument with a different norm to handle well-posedness limitations.

## Key findings

- Solutions are globally bounded in the $L^6_{t,x}$ space-time norm.
- Solutions scatter in the energy space as time approaches infinity.
- The method adapts existing techniques with a modified norm for exponential nonlinearities.

## 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 $H^1(\mathbb{R}) \times L^2(\mathbb{R})$. We prove that any energy solution has a global bound of the $L^6_{t,x}$ space-time norm, and hence scatters in $H^1(\mathbb{R}) \times L^2(\mathbb{R})$ as $t\rightarrow\pm \infty$. The proof is based on the argument by Killip-Stovall-Visan (Trans. Amer. Math. Soc. 364 (2012), no. 3, 1571--1631). However, since well-posedness in $H^{1/2}(\mathbb{R}) \times H^{-1/2}(\mathbb{R})$ for NLKG with the exponential nonlinearity holds only for small initial data, we use the $L_t^6 W^{s-1/2,6}_x$-norm for some $s>\frac{1}{2}$ instead of the $L_{t,x}^6$-norm, where $W_x^{s,p}$ denotes the $s$-th order $L^p$-based Sobolev space.