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

Abstract

We consider a class of power-type nonlinear Schrödinger 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.

62 pages