Mathématiques   > Accueil   > Avancées en Mathématiques Pures et Appliquées   > À paraître   > Article

# [Forthcoming] Théorie de diffusion dans les espaces L2 pondérés pour une classe de l’équation de Schrödinger non-linéaire inhomogène défocalisée

## [Forthcoming] Scattering theory in weighted L2 space for a class of the defocusing inhomogeneous nonlinear Schrödinger equation

VAN DUONG DINH
Université de Lille CNRS

Publié le 21 septembre 2020   DOI :

### Keywords

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 < $\alpha$ < $\alpha^\star$ 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 $\alpha_\star$ < $\alpha$ < $\alpha^\star$, where $\alpha_\star = \frac{4-2b}{d}$.

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 < $\alpha$ < $\alpha^\star$ 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 $\alpha_\star$ < $\alpha$ < $\alpha^\star$, where $\alpha_\star = \frac{4-2b}{d}$.