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Global well-posedness and scattering for nonlinear Schrödinger equations with combined nonlinearities in the radial case

Xing Cheng, Changxing Miao, Lifeng Zhao

Abstract

We consider the Cauchy problem for the nonlinear Schrödinger equation with combined nonlinearities, one of which is defocusing mass-critical and the other is focusing energy-critical or energy-subcritical. The threshold is given by means of variational argument. We establish the profile decomposition in $H^1(\Bbb R^d)$ and then utilize the concentration-compactness method to show the global wellposedness and scattering versus blowup in $H^1(\Bbb R^d)$ below the threshold for radial data when $d\leq4$.

Global well-posedness and scattering for nonlinear Schrödinger equations with combined nonlinearities in the radial case

Abstract

We consider the Cauchy problem for the nonlinear Schrödinger equation with combined nonlinearities, one of which is defocusing mass-critical and the other is focusing energy-critical or energy-subcritical. The threshold is given by means of variational argument. We establish the profile decomposition in and then utilize the concentration-compactness method to show the global wellposedness and scattering versus blowup in below the threshold for radial data when .

Paper Structure

This paper contains 7 sections, 25 theorems, 268 equations.

Table of Contents

  1. Introduction
  2. Variational estimates
  3. Wellposedness and perturbation theory
  4. Linear Profile decomposition
  5. Extraction of a critical element
  6. Extinction of the critical element
  7. Blow-up

Key Result

Proposition 1.1

For $1 + \frac{4}{d} < p < 1 + \frac{4}{d-2},\, d\ge 3, \text{ and } \ 1 + \frac{4}{d} < p < \infty,\, d= 1,2$, $\omega > 0$, we have $m_\omega > 0$. Moreover, $m_\omega = \mathcal{S}_\omega(Q)$, where $Q\in H^1(\mathbb{R}^d)$ is the ground state of

Theorems & Definitions (46)

  • Proposition 1.1
  • Proposition 1.2
  • Remark 1
  • Theorem 1.3
  • Lemma 1.4: Strichartz estimate, Keel-Tao
  • Lemma 2.1: Dynamic behavior of $\mathcal{K}$ under the scaling
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • ...and 36 more