Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Energy-critical inhomogeneous nonlinear Schrödinger equation with two power-type nonlinearities

Andressa Gomes, Mykael Cardoso

TL;DR

The paper studies the DINLS with two inhomogeneous nonlinearities, where one term is energy-critical in $\dot H^1$, and proves global well-posedness in several parameter regimes via a perturbative-in-energy framework, kinetic-energy control, and stability theory for the energy-critical INLS. It develops local well-posedness results for the DINLS, then extends them globally in settings where the energy and mass controls close, using fixed-point and perturbation arguments. In parallel, it derives finite-time blow-up criteria through virial identities in multiple parameter regimes, highlighting the delicate balance between the two nonlinearities and their inhomogeneities. The results advance the understanding of global dynamics and blow-up phenomena for INLS with dual nonlinearities, providing a blueprint for analyzing similarly structured dispersive PDEs with competing energy-critical and intercritical effects.

Abstract

We consider the initial value problem for the inhomogeneous nonlinear Schrödinger equation with double nonlinearities (DINLS) \begin{equation*} i \partial_t u + Δu = λ_1 |x|^{-b_1}|u|^{p_1}u + λ_2|x|^{-b_2}|u|^{\frac{4-2b_2}{N-2}}u, \end{equation*} where $λ_1,λ_2\in \mathbb{R}$, $3\leq N<6$ and $0<b_1,b_2<\min\{2,\frac{6-N}{2}\}$. In this paper, we establish global well-posedness results for certain parameter regimes and prove finite-time blow-up phenomena under specific conditions. Our analysis relies on stability theory, energy estimates, and virial identities adapted to the DINLS model.

Energy-critical inhomogeneous nonlinear Schrödinger equation with two power-type nonlinearities

TL;DR

The paper studies the DINLS with two inhomogeneous nonlinearities, where one term is energy-critical in , and proves global well-posedness in several parameter regimes via a perturbative-in-energy framework, kinetic-energy control, and stability theory for the energy-critical INLS. It develops local well-posedness results for the DINLS, then extends them globally in settings where the energy and mass controls close, using fixed-point and perturbation arguments. In parallel, it derives finite-time blow-up criteria through virial identities in multiple parameter regimes, highlighting the delicate balance between the two nonlinearities and their inhomogeneities. The results advance the understanding of global dynamics and blow-up phenomena for INLS with dual nonlinearities, providing a blueprint for analyzing similarly structured dispersive PDEs with competing energy-critical and intercritical effects.

Abstract

We consider the initial value problem for the inhomogeneous nonlinear Schrödinger equation with double nonlinearities (DINLS) \begin{equation*} i \partial_t u + Δu = λ_1 |x|^{-b_1}|u|^{p_1}u + λ_2|x|^{-b_2}|u|^{\frac{4-2b_2}{N-2}}u, \end{equation*} where , and . In this paper, we establish global well-posedness results for certain parameter regimes and prove finite-time blow-up phenomena under specific conditions. Our analysis relies on stability theory, energy estimates, and virial identities adapted to the DINLS model.
Paper Structure (13 sections, 19 theorems, 225 equations)

This paper contains 13 sections, 19 theorems, 225 equations.

Table of Contents

  1. Introduction
  2. Notation and Preliminaries
  3. Notation
  4. Nonlinear Estimates
  5. Energy-critical INLS
  6. Well-posedness
  7. Short-time pertubations
  8. Long-Time Perturbations
  9. Local well-posedness for DINLS
  10. Global well-posedness to INLS with two power-type nonlinearities
  11. Kinetic Energy Control
  12. Good Local Well-Posedness
  13. Finite time Blowup

Key Result

Theorem 1.1

Let $2 < N < 6$, $0 < b_1, b_2 < \min\{2 , \frac{6-N}{2}\}$, $0 < p_1 < \frac{4-2b_1}{N-2}$, and $p_2 = \frac{4 - 2 b_2}{N-2}$. Then, for all $u_0 \in H^1(\mathbb{R}^N)$, there exists a unique global solution $u \in C\left([0,\infty); H^1(\mathbb{R}^N)\right) \cap S^1(L^2, I)$ for the Cauchy proble Moreover, for all compact intervals $I$,

Theorems & Definitions (42)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • Remark 2.4
  • Lemma 2.5
  • proof
  • Lemma 2.6
  • proof
  • ...and 32 more