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Global Well-posedness and Scattering for the Focusing Energy-critical Inhomogeneous Nonlinear Schrödinger Equation with Non-radial Data

Dongjin Park

Abstract

We consider the focusing energy-critical inhomogeneous nonlinear Schrödinger equation \[ iu_t + Δu = -|x|^{-b}|u|^αu \] where $n \geq 3$, $0<b<\min(2, n/2)$, and $α=(4-2b)/(n-2)$. We prove the global well-posedness and scattering for every $n \geq 3$, $0<b<\min(2, n/2)$, and every non-radial initial data $φ\in \dot{H}^1(\mathbb{R}^n)$ by the concentration compactness arguments of Kenig and Merle (2006) as in the work of Guzman and Murphy (2021). Lorentz spaces are adopted during the development of the stability theory in critical spaces as in Killip and Visan (2013). The result improves multiple earlier works by providing a unified approach to the scattering problem, accepting both focusing and defocusing cases, and eliminating the radial-data assumption and additional restrictions to the range of $n$ and $b$.

Global Well-posedness and Scattering for the Focusing Energy-critical Inhomogeneous Nonlinear Schrödinger Equation with Non-radial Data

Abstract

We consider the focusing energy-critical inhomogeneous nonlinear Schrödinger equation where , , and . We prove the global well-posedness and scattering for every , , and every non-radial initial data by the concentration compactness arguments of Kenig and Merle (2006) as in the work of Guzman and Murphy (2021). Lorentz spaces are adopted during the development of the stability theory in critical spaces as in Killip and Visan (2013). The result improves multiple earlier works by providing a unified approach to the scattering problem, accepting both focusing and defocusing cases, and eliminating the radial-data assumption and additional restrictions to the range of and .

Paper Structure

This paper contains 9 sections, 24 theorems, 134 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Function spaces
  4. Local well-posedness and stability
  5. Case 1: high nonlinearity power ($\alpha \geq 1$)
  6. Case 2: low nonlinearity power ($\alpha \leq 1$)
  7. Variational analysis
  8. Construction of a minimal energy blow-up solution
  9. Proof of Theorem \ref{['mainthm']}

Key Result

Theorem 1.1

Let $n\geq 3$ be an integer and $b>0$ obey the following range restriction. Let $[T_-, T_+]$ be a compact interval, and let $u$ be the unique spherically symmetric solution of INLS associated with the initial data $\phi \in H^1(\mathbb{R}^n)$ and lying in the solution space where $p_* = 2(n+2)/(n-2+2b)$ and $q_* = 2n(n+2)/(n^2+4-4b)$. Then we have where $E = E[\phi]$ is the solution's energy an

Theorems & Definitions (39)

  • Theorem 1.1: Park24128202
  • Theorem 1.2: Park24128202
  • Theorem 1.3
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Proposition 2.1: Hölder's inequality; Lemarie02, ONeil63
  • Proposition 2.2: Sobolev embedding; Lemarie02
  • Proposition 2.3: Equivalent characterization of Sobolev-Lorentz spaces
  • Proposition 2.4: Strichartz estimates; KeelTao
  • ...and 29 more