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Global Well-Posedness for the Fourth-Order Nonlinear Schrödinger Equation with Potential in the Energy-Critical Case

Hikaru Nakayama

Abstract

We consider the defocusing fourth-order nonlinear Schrödinger equation with potential \[ i\partial_t u + Δ^2 u + Vu + λ|u|^{p-1}u = 0 \qquad (x \in \mathbb{R}^n,\ t \in \mathbb{R}), \] in dimensions $n \ge 5$. In the energy-critical case $p = \frac{n+4}{n-4}$, under suitable assumptions on a radial real-valued potential $V$, we prove global well-posedness for radial initial data in $H^2(\mathbb{R}^n)$. We also show that every such solution scatters in $H^2$ to a free solution of the biharmonic Schrödinger equation. The proof relies on Strichartz estimates for fourth-order Schrödinger operators with potential, equivalence of Sobolev norms associated with $Δ^2+V$ and $Δ^2$, boundedness of wave operators, perturbative stability theory, and a Morawetz-type estimate adapted to the presence of a potential. This extends earlier results for the case without potential to a class of radial potentials.

Global Well-Posedness for the Fourth-Order Nonlinear Schrödinger Equation with Potential in the Energy-Critical Case

Abstract

We consider the defocusing fourth-order nonlinear Schrödinger equation with potential in dimensions . In the energy-critical case , under suitable assumptions on a radial real-valued potential , we prove global well-posedness for radial initial data in . We also show that every such solution scatters in to a free solution of the biharmonic Schrödinger equation. The proof relies on Strichartz estimates for fourth-order Schrödinger operators with potential, equivalence of Sobolev norms associated with and , boundedness of wave operators, perturbative stability theory, and a Morawetz-type estimate adapted to the presence of a potential. This extends earlier results for the case without potential to a class of radial potentials.
Paper Structure (14 sections, 28 theorems, 296 equations)

This paper contains 14 sections, 28 theorems, 296 equations.

Table of Contents

  1. Introduction and Main Result
  2. Introduction
  3. Main result
  4. Related background
  5. Notations
  6. Equivalence of Homogeneous Sobolev Norms on $L^{p}$
  7. Strichartz Estimates and $L^{p}$-$L^{q}$ Decay Estimates
  8. Local and Global Existence in the Subcritical Case
  9. Local existence in the critical case
  10. Stability in the critical case
  11. Almost conservation laws
  12. Global existence
  13. Scattering for the critical equation
  14. Appendix

Key Result

Theorem 1.4

Let $n \geq 5$, $\lambda > 0$, and $p \in (1, \, 2^{\sharp} - 1]$. Let $V \in L^{\infty} \cap L^{\frac{n}{4}}$ be a radial real-valued potential satisfying Assumptions as:wave, as:mora, and as:equ. Then, for any radial initial data $u_{0} \in H^{2}$, there exists a unique global solution $u\in C(\ma

Theorems & Definitions (61)

  • Theorem 1.4
  • Theorem 1.5
  • Example 1.6
  • Proposition 3.2
  • Proposition 3.4
  • proof
  • Proposition 3.5
  • proof
  • Proposition 4.2
  • Corollary 4.3
  • ...and 51 more