TY - Type of reference
TI - Scattering theory in weighted L2 space for a class of the defocusing inhomogeneous nonlinear Schrödinger equation
AU - VAN DUONG DINH
AB - 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 $$$\alpha_\star = \frac{4-2b}{d}$$$.
DO - 10.21494/ISTE.OP.2021.0724
JF - Advances in Pure and Applied Mathematics
KW - Inhomogeneous nonlinear Schrödinger equation, Local well-posedness, Decay solutions, Virial identity, Scattering, Weighted L2 space, Decay solutions, Local well-posedness, Inhomogeneous nonlinear Schrödinger equation, Virial identity, Scattering, Weighted L2 space,
L1 - https://openscience.fr/IMG/pdf/iste_apam21v12n3_3.pdf
LA - en
PB - ISTE OpenScience
DA - 2021/09/6
SN - 1869-6090
TT - Théorie de diffusion dans les espaces L2 pondérés pour une classe de l’équation de Schrödinger non-linéaire inhomogène défocalisée
UR - https://openscience.fr/Scattering-theory-in-weighted-L2-space-for-a-class-of-the-defocusing
IS - Issue 4 (September 2021)
VL - 12
ER -