@ARTICLE{10.21494/ISTE.OP.2021.0724, 

TITLE={Théorie de diffusion dans les espaces L<sup>2</sup> pondérés pour une classe de l&#8217;équation de Schrödinger non-linéaire inhomogène défocalisée}, 
  
AUTHOR={VAN DUONG DINH, }, 
	
JOURNAL={Avancées en Mathématiques Pures et Appliquées}, 
	
VOLUME={12}, 

NUMBER={Numéro 3 (Septembre 2021)}, 
	
YEAR={2021}, 

URL={https://openscience.fr/Theorie-de-diffusion-dans-les-espaces-L2-ponderes-pour-une-classe-de-l-equation}, 
	
DOI={10.21494/ISTE.OP.2021.0724}, 
	
ISSN={1869-6090}, 
	
ABSTRACT={In this paper, we consider the following inhomogeneous nonlinear Schrödinger equation (INLS) 
$$$i\partial_t u + \Delta u + \mu$$$ |$$$x$$$|$$$^{-b}$$$|$$$u$$$|$$$^\alpha u = 0, \quad (t,x)\in ℝ \times ℝ^d$$$
with $$$b, \alpha$$$ > 0. First, we revisit the local well-posedness in $$$H^1(ℝ^d)$$$ for (INLS) of Guzmán [Nonlinear Anal. Real World Appl. 37 (2017), 249-286] and give an improvement of this result in the two and three spatial dimensional cases. Second, we study the decay of global solutions for the defocusing (INLS), i.e. $$$\mu=-1$$$ when 0 &lt; $$$\alpha$$$ &lt; $$$\alpha^\star$$$ where $$$\alpha^\star = \frac{4-2b}{d-2}$$$ for $$$d\geq 3$$$, and $$$\alpha^\star = \infty$$$ for $$$d=1, 2$$$
by assuming that the initial data belongs to the weighted $$$L^2$$$ space $$$\Sigma =\{u \in H^1(ℝ^d) :$$$ |$$$x$$$|$$$ u \in L^2(ℝ^d) \}$$$. Finally, we combine the local theory and the decaying property to show the scattering in $$$\Sigma$$$ for the defocusing (INLS) in the case $$$\alpha_\star$$$ &lt; $$$\alpha$$$ &lt; $$$\alpha^\star$$$, where $$$\alpha_\star = \frac{4-2b}{d}$$$.}}