Scattering theory below energy for the cubic fourth-order Schrödinger equation

Abstract

We investigate the global existence and scattering for the cubic fourth-order Schrödinger equation $i{u}_t+{\Delta }^{2}u+|u{|}^{2}u=0$ in the low regularity space $H^s({\mathbb{R}}^n)$ with $s<2$. We provide an alternative approach to obtain a new interaction Morawetz estimate and extend the range of the dimension of the interactive estimate in Pausader \cite{P08} by modifying a tensor product method appeared in \cite{CGT}. We combine interaction Morawetz estimates, energy increments for the I-method to prove the result.

33pages

References in corpus (3)

• Global wellposedness and scattering for the focusing energy-critical nonlinear Schrodinger equations of fourth order in the radial case
• 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

Cited by in corpus (5)

• Scattering theory for the defocusing fourth-order Schrödinger equation
• Global Well-posedness and scattering for fourth-order Schrödinger equations on waveguide manifolds
• Global well-posedness for a $L^2$-critical nonlinear higher-order Schrödinger equation
• On the decay property of the cubic fourth-order Schrödinger equation
• Random data theory for the cubic fourth-order nonlinear Schrödinger equation