Luca Fanelli: Energy Scattering for the defocusing nonlinear Klein-Gordon Equation with a magnetic field

Date: 2024-06-11

Time: 16:10 - 17:00

Luca Fanelli, University of the Basque Country


Let us consider the defocusing nonlinear Klein-Gordon equation \[ \partial_{tt}^2 u -\Delta u +m^2u+ u^p=0 \] in \(\mathbb{R}^{1+d}\). For \(p>1+\frac{4}{d}\), the global Cauchy-Theory with scattering in \(H^1\) is due to Ginibre-Soffer-Velo (\(d\geq3\)) and Nakanishi (\(d=1,2\)). The main ingredient for the Cauchy Theory are Strichartz estimates, while for the scattering the fundamental tool is a global space-time estimate (Morawetz) given by the nonlinear term in the equation. In particular, in low dimension \(d=1,2\), Nakanishi in 1999 found a way to obtain a Morawetz estimate by introducing some time-dependent multiplier. In this seminar, we will present a magnetic perturbation of the equation above of the form \[ (\partial_t-iA^0)^2 u- \sum_{j=1}^d(\nabla-iA^j)^2 u +m^2u + |u|^{p-1} u =0, \] where \(A^j=A^j(t,x):\mathbb R^{1+d}\to\mathbb R\), \(j=0,\dots,d\).

For the latter equation we provide a scattering result, in the same style as in Nakanishi, in the natural Sobolev space \(H^1_A\) associated to the magnetic field. The main challenge is to understand the algebraic properties of the time-dependent Morawetz multipliers and how do they interplay with the magnetic field.

The results are obtained in collaboration with V. Georgiev (University. Pisa) and S. Lucente (Univ. Bari).