Introduction to scattering for radial 3D NlKG below energy norm

Details Introduction to scattering for radial 3D NlKG below energy norm Introduction to scattering for radial 3D NlKG below energy norm

2008-09-23

arxiv.org/abs/0809.3835v1

Mathematics

Analysis of PDEs

31 pages, submitted

Scientific paper

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 to a Morawetz-type estimate and a radial Sobolev inequality we control the variation of an almost conserved quantity on arbitrary 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.

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