Titre : Energy scattering for a 2D HARTREE type INLS 

Auteurs : Tarek Saanouni
, Radhia Ghanmi,  

Revue : Advances in Pure and Applied Mathematics 

Numéro : Issue 1 (January 2024) 

Volume : 15 

Date : 2024/01/5 

DOI : 10.21494/ISTE.OP.2023.1048 

ISSN : 1869-6090 

Résumé : This paper studies the asymptotic behavior of energy solutions to the focusing non-linear generalized Hartree equation
$$$i u_t+\Delta  u=-｜x｜^{-\varrho}｜u｜^{p-2}(\mathcal J_\alpha *｜\cdot｜^{-\varrho}｜u｜^p)u,\quad \varrho>0,\quad p\geq2.$$$
Here, $$$u:=u(t,x)$$$, where the time variable is $$$t \in ℝ$$$ and the space variable is $$$x\inℝ^2$$$.
The source term is inhomogeneous because $$$\varrho > 0$$$. The convolution with the Riesz-potential $$$\mathcal J_\alpha:=C_\alpha｜\cdot｜^{\alpha-2}$$$ for certain $$$0 < \alpha < 2$$$ gives a non-local Hartree type non-linearity. Taking account of the standard scaling invariance, one considers the inter-critical regime  $$$1 + \frac{2-2\varrho + \alpha}2 < p < \infty$$$. It is the purpose to prove the scattering under the ground state threshold. This naturally extends the previous work by the first author for space dimensions greater than three (Scattering Theory for a Class of Radial Focusing Inhomogeneous Hartree Equations, Potential Anal. (2021)). The main difference is due to the Sobolev embedding in two space dimensions $$$H^1(ℝ^2)\hookrightarrow L^r(ℝ^2)$$$, for all $$$2 \leq r < \infty$$$. This makes any exponent of the source term be energy subcritical, contrarily to the case of higher dimensions. The decay of the inhomogeneous term $$$｜x｜^{-\varrho}$$$ is used to avoid any radial assumption. The proof uses the method of Dodson-Murphy based on Tao’s scattering criteria and Morawetz estimates. 

Éditeur : ISTE OpenScience