Energy scattering for radial focusing inhomogeneous bi-harmonic Schr\"odinger equations

TL;DR

This paper investigates the long-term behavior of solutions to a radial focusing inhomogeneous bi-harmonic Schrödinger equation, demonstrating scattering in specific regimes using a novel approach.

Contribution

It introduces a new method based on Dodson-Murphy's approach to establish scattering for radial solutions in focusing bi-harmonic Schrödinger equations.

Findings

Scattering established in focusing mass super-critical regime.

Results apply to both local and non-local source terms.

Uses a novel approach for the analysis.

Abstract

This note studies the asymptotic behavior of global solutions to the fourth-order Schr\"odinger equation $$i\dot u+\Delta^2 u+F(x,u)=0 .$$ Indeed, for both cases, local and non-local source term, the scattering is obtained in the focusing mass super-critical and energy sub-critical regimes, with radial setting. This work uses a new approach due to Dodson-Murphy \cite{dm}.