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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2409.10769 |
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Table of Contents:
- We consider the focusing generalized Hartree equation in $H^1(\R^3)$ with a potential, \begin{equation*} iu_t + Δu - V(x)u + (I_γ\ast |u|^p )|u|^{p-2} u=0, \end{equation*} where $I_γ= \frac{1}{|x|^{3-γ}}$, $p \geq 2$ and $γ< 3$. In this paper, we prove scattering for the generalized Hartree equation with a potential in the intercritical case assuming radial initial data. The novelty of our approach lies in the use of a general mass-potential condition, incorporating the potential V, which extends the standard mass-energy framework. To this end, we employ a simplified method inspired by Dodson and Murphy \cite{Dod-Mur}, based on Tao's scattering criteria and Morawetz estimates. This approach provides a more straightforward proof of scattering compared to the traditional concentration-compactness/rigidity method of Kenig and Merle \cite{KENIG}.