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

References in corpus (4)

• Global well-posedness and scattering for the defocusing, $L^2$-critical, nonlinear Schr{ö}dinger equation when $d=2$
• The defocusing energy-critical nonlinear Schrödinger equation in higher dimensions
• Global well-posedness and scattering for the defocusing cubic NLS in four dimensions
• Global Well-posedness and Scattering of Defocusing Energy subcritical Nonlinear Wave Equation in dimension 3 with radial data

Cited by in corpus (2)

• The radial defocusing nonlinear Schrödinger equation in three space dimensions
• The defocusing energy-supercritical NLS in four space dimensions