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Scattering for Defocusing NLS with Inhomogeneous Nonlinear Damping and Nonlinear Trapping Potential

David Lafontaine, Boris Shakarov

Abstract

We investigate an energy-subcritical defocusing nonlinear Schrödinger equation in $\mathbb R^3$ subject to a lower order nonlinear trapping potential and a spatially dependent nonlinear damping: \begin{equation*} i\partial_t u + Δu + i a(x) |u|^{2σ_2} u = |u|^{2σ_1} u + V(x)|u|^{2σ_3} u. \end{equation*} We prove that if the damping acts where $V$ induces concentration effects, i.e. where $V$ is either negative or non-repulsive, solutions are global and uniformly bounded in $H^1$, and scatter in the intercritical regime. A primary challenge arises from the spatial dependence of $a(x)$, which breaks the energy's monotonicity. Consequently, a uniform in time control of the $H^1$ norm of a solution is non-trivial and represents a new result even for $V = 0$. We overcome this issue by introducing a novel energy modified by virial argument, showing simultaneously a uniform bound on the energy and local energy decay estimates, which are subsequently upgraded to scattering via interaction Morawetz estimates.

Scattering for Defocusing NLS with Inhomogeneous Nonlinear Damping and Nonlinear Trapping Potential

Abstract

We investigate an energy-subcritical defocusing nonlinear Schrödinger equation in subject to a lower order nonlinear trapping potential and a spatially dependent nonlinear damping: \begin{equation*} i\partial_t u + Δu + i a(x) |u|^{2σ_2} u = |u|^{2σ_1} u + V(x)|u|^{2σ_3} u. \end{equation*} We prove that if the damping acts where induces concentration effects, i.e. where is either negative or non-repulsive, solutions are global and uniformly bounded in , and scatter in the intercritical regime. A primary challenge arises from the spatial dependence of , which breaks the energy's monotonicity. Consequently, a uniform in time control of the norm of a solution is non-trivial and represents a new result even for . We overcome this issue by introducing a novel energy modified by virial argument, showing simultaneously a uniform bound on the energy and local energy decay estimates, which are subsequently upgraded to scattering via interaction Morawetz estimates.
Paper Structure (7 sections, 9 theorems, 112 equations)

This paper contains 7 sections, 9 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Strichartz Estimates and local well-posedness
  4. Evolution of the mass, the energy, and the Morawetz functional
  5. Uniform bound on the energy and local energy decay
  6. Bilinear Estimates
  7. Scattering

Key Result

Theorem 1.4

Assume that and that $0 \leq a\in C^2(\mathbb R^3) \cap L^\infty(\mathbb R^3)$ and $V\in W^{1,\infty}(\mathbb R^3)$ satisfy the control Assumption assumControl, as well as Assumptions assumA and assumV. Then, for any $u_0 \in H^1(\mathbb R^3)$, there exists a unique global solution $u \in C([0,\infty), H^1(\mat

Theorems & Definitions (18)

  • Theorem 1.4
  • Definition 1.5
  • Theorem 1.6
  • Proposition 2.1
  • Proposition 2.2
  • proof
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • ...and 8 more