On scattering for the defocusing high dimensional inter-critical NLS

Abstract

In this paper, we study the critical norm conjecture for the intercritical nonlinear Schrödinger equation with critical index sc satisfying 1 2 < sc < 1 when d ≥ 5. Under the assumption of uniform boundedness of the critical norm, we prove the global well-posedness and scattering for the Cauchy problem. We follow the standard 'Concentration compactness/Rigidity method' established in [15,16], and treat three scenarios for the critical element respectively. Moreover, double Duhamel method and interaction Morawetz estimate are applied to exclude the critical element.

$H^s_x\times H^s_x$ scattering theory for a weighted gradient system of 3D radial defocusing NLS

“…About the classical results on L 2 x , Σ and H 1 x scattering theories for (1.2), to see more details, we can refer to the books [5,7,14,34] and the numerous references therein. Recently, different types of scattering results on (1.2) when d = 3 were established by many authors, we can see [1,4,10,11,13,15,17,18,19,20,21,22,23,24,25,26,27,28,35,38]. We specially point out that the H s x scattering theory for (1.2) with p = 2 is also studied by many authors.…”

Critical exponents line, scattering thoeries for a weighted gradient system of semilinear Schrodinger equations

“…To study it, the main difficulty is the lack of conservation laws beyond H 1 (R d ) Sobolev space. Recently, the results on Ḣs scattering for (1.2) are established by many authors, see [32,44,53,64,92] and the references therein. Very recently, in [4], Beceanu et.…”