Advances in Differential Equations

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

Soonsik Kwon and Tristan Roy

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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

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

First available in Project Euclid: 18 February 2016

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35Q55: NLS-like equations (nonlinear Schrödinger) [See also 37K10]


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.

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