Mathématiques > Accueil > Avancées en Mathématiques Pures et Appliquées > Numéro 1 (Janvier 2024) > Article
Tarek Saanouni
Departement of Mathematics
College of Science and Arts in Uglat Asugour
Qassim University
Buraydah
Kingdom of Saudi Arabia
Radhia Ghanmi
University of Tunis El Manar
Tunisia
Publié le 5 janvier 2024 DOI : 10.21494/ISTE.OP.2023.1048
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.
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.
Inhomogeneous Hartree equation scattering nonlinear equations
Inhomogeneous Hartree equation scattering nonlinear equations