Introduction to scattering for radial 3D NLKG below energy norm

We prove scattering for the radial nonlinear Klein-Gordon equation $\partial_{tt} u - \Delta u + u = -|u|^{p-1} u$ with $5 > p >3$ and data $(u_{0}, u_{1}) \in H^{s} \times H^{s-1}$, $1 > s > 1- \frac{(5-p)(p-3)}{2(p-1)(p-2)}$ if $4 \geq p > 3$ and $1 > s > 1 - \frac{(5-p)^{2}}{2(p-1)(6-p)}$ if $5> p \geq 4$. First we prove Strichartz-type estimates in $L_{t}^{q} L_{x}^{r}$ spaces. Then by using these decays we establish some local bounds. By combining these results with a Morawetz-type estimate and a radial Sobolev inequality we control the variation of an almost conserved quantity on arbitrarily large intervals. Once we have showed that this quantity is controlled, we prove that some of these local bounds can be upgraded to global bounds. This is enough to establish scattering. All the estimates involved require a delicate analysis due to the nature of the nonlinearity and the lack of scaling.