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

IF 1.3 4区数学 Q1 MATHEMATICS

Evolution Equations and Control Theory Pub Date : 2021-07-02 DOI:10.3934/eect.2022059

J. An, Jinmyong Kim

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引用次数: 0

摘要

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。

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

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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来源期刊

Evolution Equations and Control Theory MATHEMATICS, APPLIED-MATHEMATICS

CiteScore

3.10

自引率

6.70%

发文量

期刊介绍： EECT is primarily devoted to papers on analysis and control of infinite dimensional systems with emphasis on applications to PDE''s and FDEs. Topics include: * Modeling of physical systems as infinite-dimensional processes * Direct problems such as existence, regularity and well-posedness * Stability, long-time behavior and associated dynamical attractors * Indirect problems such as exact controllability, reachability theory and inverse problems * Optimization - including shape optimization - optimal control, game theory and calculus of variations * Well-posedness, stability and control of coupled systems with an interface. Free boundary problems and problems with moving interface(s) * Applications of the theory to physics, chemistry, engineering, economics, medicine and biology