Kwon
- Tristan Roy
title: Generation of decay estimates and application to scattering of rough solutions of $3D$ NLKG
---

Introduction
============

In this paper we consider the $p$- defocusing Klein-Gordon equation on $\mathbb{R}^{3}$

$$\begin{array}{ll}
\partial_{tt} u  - \Delta u + u & =  -|u|^{p-1}u \\
\end{array}
\label{Eqn:NlkgWdat}$$

with data $u(0)=u_{0}$, $\partial_{t}u(0)=u_{1}$ lying in $H^{s}$, $H^{s-1}$ respectively. Here $H^{s}$ is the standard inhomogeneous Sobolev space i.e $H^{s}$ is the completion of the Schwartz space $\mathcal{S}(\mathbb{R}^{3})$ with respect to the norm

$$\begin{array}{ll}
\| f \|_{H^{s}} & :=  \| \langle D \rangle^{s} f \|_{L^{2}(\mathbb{R}^{3})}
\end{array}$$

where $\langle D \rangle$ is the operator defined by

$$\begin{array}{ll}
\widehat{\langle D \rangle^{s} f}(\xi) & := ( 1 + |\xi|)^{s} \hat{f}(\xi)
\end{array}$$

and $\hat{f}$ denotes the Fourier transform $$\begin{array}{ll}
\hat{f}(\xi) & := \int_{\mathbb{R}^{3}} f(x) e^{-i x \cdot \xi} \, dx
\end{array}$$ We are interested in the strong solutions of the $p$- defocusing Klein-Gordon equation on some interval $[0,T]$ i.e maps $u$, $\partial_{t} u$ that lie in $C \left([0, \, T], \, H^{s}(\mathbb{R}^{3}) \right)$, $C \left( [0, \,T], \, H^{s-1} ( \mathbb{R}^{3}) \right)$ respectively and that satisfy

$$\begin{array}{ll}
u(t) & = \cos{(t \langle D \rangle)} u_{0} + \frac{\sin(t \langle D \rangle)}{\langle D \rangle} u_{1} - \int_{0}^{t} \frac{\sin \left(
