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Profile decomposition and scattering for general nonlinear Schr{ö}dinger equations

Thomas Duyckaerts, Phan van Tin

Abstract

We consider a Schr{ö}dinger equation with a nonlinearity which is a general perturbation of a power'' nonlinearity. We construct a profile decomposition adapted to this nonlinearity.We also prove global existence and scattering in a general defocusing setting, assuming thatthe critical Sobolev norm is bounded in the energy-supercritical case. This generalizes severalprevious works on double-power nonlinearities.

Profile decomposition and scattering for general nonlinear Schr{ö}dinger equations

Abstract

We consider a Schr{ö}dinger equation with a nonlinearity which is a general perturbation of a power'' nonlinearity. We construct a profile decomposition adapted to this nonlinearity.We also prove global existence and scattering in a general defocusing setting, assuming thatthe critical Sobolev norm is bounded in the energy-supercritical case. This generalizes severalprevious works on double-power nonlinearities.
Paper Structure (20 sections, 36 theorems, 311 equations)

This paper contains 20 sections, 36 theorems, 311 equations.

Table of Contents

  1. Introduction
  2. General setting
  3. Well-posedness and profile decomposition
  4. Global well-posedness
  5. Scattering
  6. Previous works
  7. Outline
  8. Preliminaries and Cauchy theory
  9. Notations and function spaces
  10. Preliminary nonlinear estimates
  11. Local Lipschitz continuity of the nonlinearity
  12. Cauchy and stability theory for general nonlinearities
  13. Profile decomposition
  14. Profiles in homogeneous Sobolev spaces
  15. Profiles in non-homogeneous Sobolev spaces
  16. ...and 5 more sections

Key Result

Theorem 1.2

Let $\iota_0$, $s_0$, $g$ such that Assumption Assum:profile holds, and such that Property Proper:bnd is valid for some $A_0\in (0,\infty]$. Let $u$ be a solution of NLS_g, with initial data in $H^{s_0}$, such that bound_Hs0 holds. Then $T_{+}(u)=+\infty$.

Theorems & Definitions (82)

  • Theorem 1.2
  • Corollary 1.3
  • Definition 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 1.8
  • Remark 1.9
  • Lemma 2.1
  • proof
  • ...and 72 more