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On the Cauchy problem for the inhomogeneous nonlinear Schrödinger equation with inverse-power potential (2406.16365v1)

Published 24 Jun 2024 in math.AP

Abstract: In this paper, we study the Cauchy problem for the inhomogeneous nonlinear Schr\"{o}dinger equation with inverse-power potential [iu_{t} +\Delta u-c|x|{-a}u=\pm |x|{-b} |u|{\sigma } u,\;\;(t,x)\in \mathbb R\times\mathbb R{d},] where $d\in \mathbb N$, $c\in \mathbb R$, $a,b>0$ and $\sigma>0$. First, we establish the local well-posedness in the fractional Sobolev spaces $Hs(\mathbb Rd)$ with $s\ge 0$ by using contraction mapping principle based on the Strichartz estimates in Sobolev-Lorentz spaces. Next, the global existence and blow-up of $H1$-solution are investigated. Our results extend the known results in several directions.

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