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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2409.10769 |
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| _version_ | 1866916397186023424 |
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| author | Guzmán, Carlos M. Loli, Cristian Yapu, Luis P. |
| author_facet | Guzmán, Carlos M. Loli, Cristian Yapu, Luis P. |
| 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}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_10769 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Scattering for the generalized Hartree equation with a potential Guzmán, Carlos M. Loli, Cristian Yapu, Luis P. Analysis of PDEs 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}. |
| title | Scattering for the generalized Hartree equation with a potential |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2409.10769 |