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Energy scattering for the unsteady damped nonlinear Schrodinger equation

Makram Hamouda, Mohamed Majdoub

TL;DR

This work analyzes the long-time dynamics of the focusing nonlinear Schrödinger equation with time-dependent damping in the energy-subcritical to intercritical range. It develops a damping-augmented framework, including the transform $v=e^{\mathbf A(t)}u$ and energy identities, together with Strichartz estimates and Grönwall-type bootstraps to establish uniform $H^1$ control and scattering. The main contributions are: (i) $H^1$-scattering for $1<p\le 1+\frac{4}{N}$ under $\underline{\mathbf a}>0$ (Theorem scat1); (ii) global existence and scattering for $1+\frac{4}{N}<p<1+\frac{4}{(N-2)_+}$ under a small-damped-Strichartz condition, plus a conditional result when $\sup_{t\ge0}\|u(t)\|_{L^{p+1}}<\infty$; and (iii) a scattering result under an exponential decay bound on the gradient (Theorem scat3-bis). The results extend constant-damping scattering theory to time-dependent damping and clarify how damping strength interacts with initial data to yield global behavior. Open questions include scattering with $\underline{\mathbf a}=0$ and extensions to slowly decaying damping and energy-critical regimes.

Abstract

We investigate the large time behavior of the solutions to the nonlinear focusing Schrödinger equation with a time-dependent damping in the energy sub-critical regime. Under non classical assumptions on the unsteady damping term, we prove some scattering results in the energy space.

Energy scattering for the unsteady damped nonlinear Schrodinger equation

TL;DR

This work analyzes the long-time dynamics of the focusing nonlinear Schrödinger equation with time-dependent damping in the energy-subcritical to intercritical range. It develops a damping-augmented framework, including the transform and energy identities, together with Strichartz estimates and Grönwall-type bootstraps to establish uniform control and scattering. The main contributions are: (i) -scattering for under (Theorem scat1); (ii) global existence and scattering for under a small-damped-Strichartz condition, plus a conditional result when ; and (iii) a scattering result under an exponential decay bound on the gradient (Theorem scat3-bis). The results extend constant-damping scattering theory to time-dependent damping and clarify how damping strength interacts with initial data to yield global behavior. Open questions include scattering with and extensions to slowly decaying damping and energy-critical regimes.

Abstract

We investigate the large time behavior of the solutions to the nonlinear focusing Schrödinger equation with a time-dependent damping in the energy sub-critical regime. Under non classical assumptions on the unsteady damping term, we prove some scattering results in the energy space.
Paper Structure (8 sections, 14 theorems, 81 equations)

This paper contains 8 sections, 14 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. Main results
  3. Useful tools & Auxiliary results
  4. Proofs of main results
  5. Proof of Theorem \ref{['scat1']}
  6. Proof of Theorem \ref{['scat3']}
  7. Proof of Theorem \ref{['scat3-bis']}
  8. Conclusion and Open Problems

Key Result

Theorem 2.1

(Cazenave) Let $N\geq 1$ and $\mathbf a(t)\equiv 0$. Assume that $1<p<1+\frac{4}{(N-2)_+}$. If $u_0\in H^1(\mathbb R^N)$ then there exist $T_{max}=T_{max}(\|u_0\|_{H^1}, N, p)>0$ and a unique maximal solution $u$ of main such that for any admissible pair $(q,r)$ in the sense of Definition Admis below. Moreover, we have the following blow-up criterion:

Theorems & Definitions (32)

  • Definition 2.1
  • Remark 2.1
  • Theorem 2.1
  • Remark 2.2
  • Theorem 2.2
  • Remark 2.3
  • Theorem 2.3
  • Remark 2.4
  • Theorem 2.4
  • Remark 2.5
  • ...and 22 more