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arXiv · 2412.00448

Blow up versus scattering below the mass-energy threshold for the focusing NLH with potential

Shuang Ji·Jing Lu
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Abstract

In this paper, we study the blow up and scattering result of the solution to the focusing nonlinear Hartree equation with potential $$i\partial_t u +\Delta u - Vu = - (|\cdot|^{-3} \ast |u|^2)u, \qquad (t, x) \in \mathbb{R} \times \mathbb{R}^5 $$ in the energy space ${H}^1(\mathbb{R}^5)$ below the mass-energy threshold. The potential $V$ we considered is an extension of Kato potential in some sense. We extend the results of Meng [26] to nonlinear Hartree equation with potential $V$ under some conditions. By establishing a Virial-Morawetz estimate and a scattering criteria, we obtain the scattering theory based on the method from Dodson-Murphy [11].

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Shuang Ji, Jing Lu. 2024-11-30. Blow up versus scattering below the mass-energy threshold for the focusing NLH with potential. https://arxiv.org/abs/2412.00448

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