Scattering for radial bi-harmonic Hartree equations

TL;DR

This paper investigates the long-term behavior of solutions to a fourth-order Hartree equation, demonstrating scattering phenomena in specific regimes for radial solutions, applicable to both attractive and repulsive interactions.

Contribution

It establishes scattering results for radial solutions of the fourth-order Hartree equation in mass super-critical and energy sub-critical regimes, expanding understanding of such equations.

Findings

Scattering proven for both attractive and repulsive cases.

Results valid in mass super-critical and energy sub-critical regimes.

Analysis focused on radial solutions.

Abstract

This note studies the asymptotic behavior of global solutions to the fourth-order generalized Hartree equation $$i\dot u+\Delta^2 u\pm(I_\alpha*|u|^p)|u|^{p-2}u=0.$$ Indeed, for both attractive and repulsive sign, the scattering is obtained in the mass super-critical and energy sub-critical regimes, with radial setting.