Scattering for a class of inhomogeneous generalized Hartree equations

ABSTRACT

This paper studies the asymptotic behavior of energy solutions to a non-linear generalized Hartree equation. Indeed, in the inter-critical regime, one revisits the scattering versus finite time blow-up of energy solutions with non-necessarily spherically symmetric datum. Here, one uses the new approach due to Dodson and Murphy. The novelty in this work is to express the scattering threshold in terms of some non-conserved quantities. The main result of this note seems to be stronger than the classical scattering versus finite time blow-up dichotomy given in terms of the conserved mass and energy by Holmer and Roudenko. Indeed, as an application, one investigates the scattering in three different regimes: under, at and beyond the ground state threshold. The main result given here, which can be seen as a criteria of scattering versus finite time blow-up of energy solutions, enables us to give a unified approach to deal with the above generalised Hartree equation in different regimes. This paper follows some new ideas presented recently in the classical Schrödinger equation with a local source term, by V.D. Dinh.

COMMUNICATED BY:

KEYWORDS:

• Inhomogeneous Hartree equation
• energy
• scattering
• blow-up
• Nonlinear equations

MATHS:

• 35Q55

Disclosure statement

No potential conflict of interest was reported by the author(s).

Data availability statement

The data that support the findings of this study are available from the corresponding author upon reasonable request.

Additional information

Funding

This work is partially supported by NSF of Gansu Province (China) (Nos. 20JR5RA498, 21JR7RE176) and Innovation Ability Promotion Foundation of Universities in Gansu Province (No. 2020B-185).