On the decay property of the cubic fourth-order Schrödinger equation

Abstract

In this short paper, we prove that the solution of the cubic fourth-order Schrödinger equation (4NLS) on ${\mathbb{R}}^d$ ($5\le d\le 8$) enjoys the same (pointwise) decay property as its linear solution does. This result is proved via a bootstrap argument based on the corresponding global result Pausader \cite{Pau1}. This result can be extended to more general dispersive equations (including some more 4NLS models) with scattering asymptotics.

arXiv comment10 pages. Comments are welcome!

References in corpus (9)

• Global wellposedness and scattering for the focusing energy-critical nonlinear Schrodinger equations of fourth order in the radial case
• Global well-posedness of the energy-critical defocusing NLS on $\mathbb{R}\times {\mathbb{T}}^3$
• Global wellposedness and scattering for the defocusing energy-critical nonlinear Schrodinger equations of fourth order in dimensions $d\ge 9$
• Wellposedness of Cauchy problem for the Fourth Order Nonlinear Schrödinger Equations in Multi-dimensional Spaces
• Scattering for nonlinear Schrodinger equation under partial harmonic confinement
• Scattering theory below energy for the cubic fourth-order Schrödinger equation
• Scattering theory for the defocusing fourth-order Schrödinger equation
• On scattering for the defocusing quintic nonlinear Schrödinger equation on the two-dimensional cylinder
• Global well-posedness and scattering for mass-critical, defocusing, infinite dimensional vector-valued resonant nonlinear Schrödinger system