Global well-posedness and scattering for the focusing energy-critical nonlinear Schrödinger equations of fourth order in the radial case

DOI10.1016/j.jde.2008.11.011zbMath1173.35117arXiv0807.0690OpenAlexW2077663392MaRDI QIDQ1018383

Lifeng Zhao, Gui Xiang Xu, Chang Xing Miao

Publication date: 19 May 2009

Published in: Journal of Differential Equations (Search for Journal in Brave)

Full work available at URL: https://arxiv.org/abs/0807.0690

zbMATH Keywords

scatteringglobal well-posednessfocusingradialenergy-criticalfourth-order Schrödinger equations

Mathematics Subject Classification ID

NLS equations (nonlinear Schrödinger equations) (35Q55) Quantum scattering theory (81U99)

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Cites Work

• Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation
• The focusing energy-critical fourth-order Schrödinger equation with radial data
• Global well-posedness, scattering and blow-up for the energy-critical, focusing, nonlinear Schrö\-dinger equation in the radial case
• Well-posedness of Cauchy problem for the fourth order nonlinear Schrödinger equations in multi-dimensional spaces
• Global well-posedness for energy critical fourth-order Schrödinger equations in the radial case
• Scattering for the nonradial 3D cubic nonlinear Schrödinger equation
• The cubic nonlinear Schrödinger equation in two dimensions with radial data
• Stability of solitons described by nonlinear Schrödinger-type equations with higher-order dispersion
• The global Cauchy problem and scattering of solutions for nonlinear Schrödinger equations in \(H^s\).
• The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions
• The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher
• Global well-posedness and scattering for the energy-critical Schrödinger equation in \(\mathbb R^{3}\)
• Wellposedness for the fourth order nonlinear Schrödinger equations
• Global well-posedness and scattering for the higher-dimensional energy-critical nonlinear Schrö\-dinger equation for radial data
• Global well-posedness of the Cauchy problem for nonlinear Schrödinger-type equations
• Critical exponents, critical dimensions and the biharmonic operator
• Self-Focusing with Fourth-Order Dispersion
• Modified wave operators for the fourth-order non-linear Schrödinger-type equation with cubic non-linearity
• The cauchy problem for the critical nonlinear Schrödinger equation in Hs
• Minimal-mass blowup solutions of the mass-critical NLS
• Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in R 1+4
• Scattering for 𝐻̇^{1/2} bounded solutions to the cubic, defocusing NLS in 3 dimensions
• The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher
• Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case
• High Frequency Approximation of Solutions to Critical Nonlinear Wave Equations
• Dispersion estimates for fourth order Schrödinger equations
• On the defect of compactness for the Strichartz estimates of the Schrödinger equations