THE MASS-CRITICAL FOURTH-ORDER SCHRÖDINGER EQUATION IN HIGH DIMENSIONS

DOI10.1142/S0219891610002256zbMath1232.35156arXiv0906.1547OpenAlexW2035102641WikidataQ115245249 ScholiaQ115245249MaRDI QIDQ3067124

Shuanglin Shao, Benoit Pausader

Publication date: 20 January 2011

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

Full work available at URL: https://arxiv.org/abs/0906.1547

zbMATH Keywords

scatteringglobal well-posednessfourth-order Schrödinger equationfourth-order dispersive equations

Mathematics Subject Classification ID

NLS equations (nonlinear Schrödinger equations) (35Q55) PDEs in connection with quantum mechanics (35Q40) Lasers, masers, optical bistability, nonlinear optics (78A60) Initial value problems for nonlinear higher-order PDEs (35G25)

Related Items (46)

On the decay property of the cubic fourth-order Schrödinger equation ⋮ The asymptotic property for nonlinear fourth-order Schrödinger equation with gain or loss ⋮ Asymptotics for the fourth-order nonlinear Schrödinger equation in 2D ⋮ Unconditional uniqueness of solution for \(\dot{H}^{s_c}\) critical 4th order NLS in high dimensions ⋮ On the inhomogeneous fourth-order nonlinear Schrödinger equation ⋮ Random data theory for the cubic fourth-order nonlinear Schrödinger equation ⋮ Global well-posedness for a \(L^2\)-critical nonlinear higher-order Schrödinger equation ⋮ Higher-order nonlinear Schrödinger equation in 2D case ⋮ Global well-posedness and scattering for the nonstandard defocusing beam equation ⋮ Decay in energy space for the solution of fourth-order Hartree-Fock equations with general non-local interactions ⋮ Global well-posedness and scattering of a generalized nonlinear fourth-order wave equation ⋮ Local well-posedness of a critical inhomogeneous bi-harmonic Schrödinger equation ⋮ Factorization technique for the fourth-order nonlinear Schrödinger equation ⋮ The existence and stability of normalized solutions for a bi-harmonic nonlinear Schrödinger equation with mixed dispersion ⋮ Sharp well-posedness of the Cauchy problem for the fourth order nonlinear Schrödinger equation ⋮ Nonlinear fourth-order Schrödinger equations with radial data ⋮ A refinement of the Strichartz inequality for the wave equation with applications ⋮ Global Well-Posedness and Scattering for Fourth-Order Schrödinger Equations on Waveguide Manifolds ⋮ Unnamed Item ⋮ A weak form of the soliton resolution conjecture for high-dimensional fourth-order Schrödinger equations ⋮ Large time asymptotics for the fourth-order nonlinear Schrödinger equation ⋮ Maximizers for Gagliardo-Nirenberg inequalities and related non-local problems ⋮ Lower regularity solutions of the biharmonic Schrödinger equation in a quarter plane ⋮ Large time asymptotic behavior of solutions to higher order nonlinear Schrödinger equation ⋮ Smoothing property for the \( L^2 \)-critical high-order NLS. II ⋮ Dynamics of radial solutions for the focusing fourth-order nonlinear Schrödinger equations ⋮ Scattering operator for the fourth order nonlinear Schrödinger equation ⋮ Scattering theory for the defocusing fourth order NLS with potentials ⋮ Decay of solutions to a fourth-order nonlinear Schrödinger equation ⋮ Global existence and scattering for a class of nonlinear fourth-order Schrödinger equation below the energy space ⋮ Global existence and asymptotic behavior of solutions to the fourth-order nonlinear Schrödinger equation in the critical case ⋮ The linear profile decomposition for the fourth order Schrödinger equation ⋮ Dispersive estimates for inhomogeneous fourth-order Schrödinger operator in 3D with zero energy obstructions ⋮ Orbitally Stable Standing Waves of a Mixed Dispersion Nonlinear Schrödinger Equation ⋮ Scattering of solutions to the fourth-order nonlinear Schrödinger equation ⋮ Unconditional uniqueness of higher order nonlinear Schrödinger equations ⋮ Asymptotics for the fourth-order nonlinear Schrödinger equation in the critical case ⋮ Scattering for the focusingL²-supercritical andḢ²-subcritical biharmonic NLS equations ⋮ Normalized solutions to the mixed dispersion nonlinear Schrödinger equation in the mass critical and supercritical regime ⋮ On a fourth‐order nonlinear Helmholtz equation ⋮ The initial-boundary value problem for the biharmonic Schrödinger equation on the half-line ⋮ \(H^2\)-scattering for systems of weakly coupled Fourth-order NLS equations in low space Dimensions ⋮ Asymptotics for a class of bi-harmonic coupled Schrödinger equations ⋮ Dissipative property for higher order nonlinear Schrödinger equation ⋮ Probabilistic Cauchy theory for the mass-critical fourth-order nonlinear Schrödinger equation ⋮ The sharp Strichartz and Sobolev-Strichartz inequalities for the fourth-order Schrödinger equation

Cites Work

• Unnamed Item
• The linear profile decomposition for the fourth order Schrödinger equation
• Global well-posedness, scattering and blow-up for the energy-critical focusing non-linear wave equation
• Nonlinear Schrödinger equations and sharp interpolation estimates
• The focusing energy-critical fourth-order Schrödinger equation with radial data
• Dynamic of threshold solutions for energy-critical NLS
• Global well-posedness, scattering and blow-up for the energy-critical, focusing, nonlinear Schrö\-dinger equation in the radial case
• Global well-posedness for energy critical fourth-order Schrödinger equations in the radial case
• A global compact attractor for high-dimensional defocusing nonlinear Schrödinger equations with potential
• Scattering for the nonradial 3D cubic nonlinear Schrödinger equation
• The cubic fourth-order Schrödinger equation
• Global well-posedness and scattering for the focusing energy-critical nonlinear Schrödinger equations of fourth order in the radial case
• Determination of blow-up solutions with minimal mass for nonlinear Schrödinger equations with critical power
• Stability of solitons described by nonlinear Schrödinger-type equations with higher-order dispersion
• Trilinear resonant interactions of semilinear hyperbolic waves
• The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions
• The blow-up dynamic and upper bound on the blow-up rate for critical nonlinear Schrödinger equation
• Global well-posedness and scattering for the energy-critical Schrödinger equation in \(\mathbb R^{3}\)
• Scattering and the Levandosky-Strauss conjecture for fourth-order nonlinear wave equations
• Global well-posedness and scattering for the defocusing mass-critical nonlinear Schrödinger equation for radial data in high dimensions
• A (concentration-)compact attractor for high-dimensional nonlinear Schrödinger equations
• The Cauchy problem for the fourth-order nonlinear Schrödinger equation related to the vortex filament
• On the role of quadratic oscillations in nonlinear Schrödinger equations II. The $L^2$-critical case
• Global well-posedness and scattering for the defocusing energy-critical nonlinear Schrödinger equation in R 1+4
• Blow up of the critical norm for some radial <i>L</i><sup xmlns:m="http://www.w3.org/1998/Math/MathML" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> super critical nonlinear Schrödinger equations
• Scattering for 𝐻̇^{1/2} bounded solutions to the cubic, defocusing NLS in 3 dimensions
• Mass concentration phenomena for the $L^2$-critical nonlinear Schrödinger equation
• Endpoint Strichartz estimates
• Description of the lack of compactness for the Sobolev imbedding
• Global wellposedness of defocusing critical nonlinear Schrödinger equation in the radial case
• High Frequency Approximation of Solutions to Critical Nonlinear Wave Equations
• Motion and expansion of a viscous vortex ring. Part 1. A higher-order asymptotic formula for the velocity
• Remark on well-posedness for the fourth order nonlinear Schrödinger type equation
• A Refined Well-Posedness for the Fourth-Order Nonlinear Schrödinger Equation Related to the Vortex Filament
• On a sharp lower bound on the blow-up rate for the 𝐿² critical nonlinear Schrödinger equation
• On the defect of compactness for the Strichartz estimates of the Schrödinger equations

This page was built for publication: THE MASS-CRITICAL FOURTH-ORDER SCHRÖDINGER EQUATION IN HIGH DIMENSIONS