---
abstract: |
    We prove scattering of solutions just below the energy norm of the $3D$ Klein-Gordon equation

    $$\left\{
    \begin{array}{ccl}
    \partial_{tt} u  - \Delta u + u & = & -|u|^{p-1} u \\
    u(0,x)& = & u_{0}(x)    \\
    \partial_{t} u(0,x) & = & u_{1}(x)
    \end{array}
    \right. \nonumber$$

    with $5 > p >3$: this result extends those obtained in the energy class [@brenscat; @nakanash1; @nakanash2] and those obtained below the energy norm under the additional assumption of spherical symmetry [@triroyradnlkg]. It is well-known that the notion of scattering is closely related to that of decay. The main difficulty is that, unlike the radial case, there is no useful decay estimate available. The idea is to generate some decay estimates, by means of concentration [@bourgjams] and a low-high frequency decomposition [@bourgbook; @almckstt]. On low frequencies we use the physical properties of the solution and we modify the arguments in [@nakanash1; @nakanash2]; on high frequencies we use the local smoothing properties of the solutions: this, combined with an almost conservation law, allows to prove these decay estimates: this is the starting point to establish scattering.
address:
- Korea Advanced Institute Of Science And Technology 
- 'Institute For Advanced Study, Princeton'
author:
- Soonsik 