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

TL;DR

This paper proves that solutions to a class of inter-critical nonlinear Schrödinger equations are global and scatter if they remain bounded in the critical Sobolev space, extending understanding of solution behavior in this regime.

Contribution

It establishes a scattering criterion based on boundedness in the critical Sobolev space for inter-critical NLS equations, using concentration-compactness methods.

Findings

Solutions bounded in the critical Sobolev space are global.

Bounded solutions in the critical space scatter as time goes to infinity.

The result applies to power-type nonlinearities between mass- and energy-critical exponents.

Abstract

We consider a class of power-type nonlinear Schr\"odinger equations for which the power of the nonlinearity lies between the mass- and energy-critical exponents. Following the concentration-compactness approach, we prove that if a solution $u$ is bounded in the critical Sobolev space throughout its lifespan, that is, $u\in L_t^\infty \dot{H}_x^{s_c}$, then $u$ is global and scatters.