# Talk by Luca Fanelli

On May 2nd, 2024, **Luca Fanelli **(Ikerbasque, Bilbao) gave a talk about **"Scattering in the energy space for the 2D defocusing nonlinear Klein-Gordon equation with a magnetic field"** as part of the research seminar Analysis of the FernUniversität in Hagen. This lecture is partially supported by the COST action Mathematical models for interacting dynamics on networks.

## Abstract

Let us consider the defocusing nonlinear Klein-Gordon equation

(1) −□um^{2}uu^{p} = 0

in R^{1d} where □ = −(∂_{tt})^{2}∆. For p ≥ 1-4/d, the global Cauchy-Theory with scattering in H^{1} is due to Ginibre-Soffer-Velo (d ≥ 3) 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 equation (1) of the form

(2) −□_{A}um^{2}u|u|^{p-1}u = 0,

where

□_{A} = −(∂t − iA^{0})^{2} Σ_{j=1,...,d} (∇ − iA^{j})^{2} . A^{j} = A^{j}(t, x) : R^{1d} → R, j = 0, . . . , d.

For equation (2), 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 (University Bari).