## Advances in Differential Equations

### Scattering of rough solutions of the nonlinear Klein-Gordon equations in 3D

#### Abstract

We prove scattering of solutions below the energy norm of the nonlinear Klein-Gordon equation in 3D with a defocusing power-type nonlinearity that is superconformal and energy subcritical: this result extends those obtained in the energy class [4, 18, 19] and those obtained below the energy norm under the additional assumption of spherical symmetry [25]. In order to do that, we generate an exponential-type decay estimate in $H^{s}$, $s < 1$, by means of concentration [1] and a low-high frequency decomposition [2, 7]: this is the starting point to prove scattering. On low frequencies, we modify the arguments in [18, 19]; on high frequencies, we use the smoothing effect of the solutions to control the error terms: this, combined with an almost conservation law, allows to prove this decay estimate.

#### Article information

Source
Adv. Differential Equations, Volume 21, Number 3/4 (2016), 333-372.

Dates
First available in Project Euclid: 18 February 2016

Permanent link to this document
https://projecteuclid.org/euclid.ade/1455805261

Mathematical Reviews number (MathSciNet)
MR3461297

Zentralblatt MATH identifier
1337.35135

#### Citation

Kwon, Soonsik; Roy, Tristan. Scattering of rough solutions of the nonlinear Klein-Gordon equations in 3D. Adv. Differential Equations 21 (2016), no. 3/4, 333--372. https://projecteuclid.org/euclid.ade/1455805261