The Cauchy problem for the critical inhomogeneous nonlinear Schrödinger equation in $ H^{s}(\mathbb R^{n}) $

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2021-07-02 DOI:10.3934/eect.2022059

J. An, Jinmyong Kim

引用次数: 0

Abstract

In this paper, we study the Cauchy problem for the critical inhomogeneous nonlinear Schrödinger (INLS) equation iut +∆u = |x| f(u), u(0) = u0 ∈ H (R), where n ≥ 3, 1 ≤ s < n2 , 0 < b < 2 and f(u) is a nonlinear function that behaves like λ |u| σ u with λ ∈ C and σ = 4−2b n−2s . We establish the local well-posedness as well as the small data global well-posedness and scattering in H(R) with 1 ≤ s < n2 for the critical INLS equation under some assumption on b. To this end, we first establish various nonlinear estimates by using fractional Hardy inequality and then use the contraction mapping principle based on Strichartz estimates.

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H^{s}(\mathbb R^{n}) $中临界非齐次非线性Schrödinger方程的Cauchy问题

本文研究了临界非齐次非线性Schrödinger (INLS)方程iut +∆u = |x| f(u)， u(0) = u0∈H (R)的Cauchy问题，其中n≥3,1≤s < n2, 0 < b < 2，且f(u)是λ∈C， σ = 4−2b n−2s时表现为λ |u| σ u的非线性函数。对于临界INLS方程，在b上的某些假设条件下，我们建立了局部适定性和小数据全局适定性以及1≤s < n2时在H(R)上的散射。为此，我们首先利用分数阶Hardy不等式建立了各种非线性估计，然后利用基于Strichartz估计的收缩映射原理。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

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