Energy-supercritical NLS: critical ${\dot{H}}^s$-bounds imply scattering

Abstract

We consider two classes of defocusing energy-supercritical nonlinear Schrödinger equations in dimensions $d\ge 5$. We prove that if the solution $u$ is apriorily bounded in the critical Sobolev space, that is, $u\in L_t^{\infty }{\dot{H}}_x^{s_c}$, then $u$ is global and scatters.

References in corpus (6)

• The mass-critical nonlinear Schrödinger equation with radial data in dimensions three and higher
• The nonlinear Schrödinger equation with combined power-type nonlinearities
• The cubic nonlinear Schrödinger equation in two dimensions with radial data
• The focusing energy-critical nonlinear Schrödinger equation in dimensions five and higher
• Maximizers for the Strichartz inequalities and the Sobolev-Strichartz inequalities for the Schrödinger equation
• Scattering for H^1/2 bounded solutions to the cubic, defocusing NLS in 3 dimensions

Cited by in corpus (6)

• Going beyond the threshold: scattering and blow-up in the focusing NLS equation
• Scattering for radial, semi-linear, super-critical wave equations with bounded critical norm
• Global well-posedness and scattering for the defocusing cubic NLS in four dimensions
• Numerical simulations of the energy- supercritical Nonlinear Schrödinger equation
• Strichartz type Inequalities for Parabolic and Schrödinger Equations in rearrangement invariant Spaces
• Scattering theory for energy-supercritical Klein-Gordon equation