Scattering theory in a weighted $L^2$ space for a class of the defocusing inhomogeneous nonlinear Schrödinger equation

Abstract

In this paper, we consider the following inhomogeneous nonlinear Schrödinger equation (INLS) \[ i\partial_t u + Δu + μ|x|^{-b} |u|^αu = 0, \quad (t,x)\in \mathbb{R} \times \mathbb{R}^d \] with $b,\alpha >0$. First, we revisit the local well-posedness in $H^1({\mathbb{R}}^d)$ 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 $d\ge 3$, and ${\alpha }^{\star }=\infty$ for $d=1,2$ by assuming that the initial data belongs to the weighted $L^2$ space $\Sigma =\{u\in H^1({\mathbb{R}}^d):|x|u\in L^2({\mathbb{R}}^d)\}$. 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 ${\alpha }_{\star }=\frac{4-2b}{d}$.

26 pages, accepted to Advances in Pure and Applied Mathematics

References in corpus (2)

• Scattering for the radial focusing INLS equation in higher dimensions
• Energy scattering for a class of the defocusing inhomogeneous nonlinear Schrödinger equation

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• Blow-up of radial solutions for the intercritical inhomogeneous NLS equation
• A remark on the scattering theory for the 2d radial focusing INLS
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• The Cauchy problem for the energy-critical inhomogeneous nonlinear Schrödinger equation
• Blowup of $H^1$ solutions for a class of the focusing inhomogeneous nonlinear Schrödinger equation
• On well-posedness for the inhomogeneous nonlinear Schrödinger equation in the critical case
• Bilinear Strichartz's type estimates in Besov spaces with application to inhomogeneous nonlinear biharmonic Schrödinger equation
• Scattering and blow-up criteria for 3D cubic focusing nonlinear inhomogeneous NLS with a potential
• Long time behaviors for the inhomogeneous NLS with a potential in ${\mathbb{R}}^3$
• Local and global well-posedness in $L^2({\mathbb{R}}^n)$ for the inhomogeneous nonlinear Schrödinger equation