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Research Article | Open Access

On the nonlinear Schrödinger equation with critical source term: global well-posedness, scattering and finite time blowup

Saleh Almuthaybiri1, Radhia Ghanmi2, Tarek Saanouni1( )
Department of Mathematics, College of Science, Qassim University, Saudi Arabia
University of Tunis El Manar, Faculty of Sciences of Tunis, 2092 Tunis, LR03ES04 Partial Differential Equations, Tunisia
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An erratum to this article is available online at:
https://doi.org/10.3934/math.2025112

Abstract

This study explored the time asymptotic behavior of the Schrödinger equation with an inhomogeneous energy-critical nonlinearity. The approach follows the concentration-compactness method due to Kenig and Merle. To address the primary challenge posed by the singular inhomogeneous term, we utilized Caffarelli-Kohn-Nirenberg weighted inequalities. This work notably expanded the existing literature by applying these techniques to higher spatial dimensions without requiring any spherically symmetric assumption.

Keywords

Schrödinger problemenergy-criticalnonlinear equationsfixed point methodglobal existencescatteringblowup
CLC number: 35Q55

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AIMS Mathematics
Volume 9 Issue 11,
November 2024
Pages 30230-30262
DOI: 10.3934/math.20241460

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Almuthaybiri S, Ghanmi R, Saanouni T. On the nonlinear Schrödinger equation with critical source term: global well-posedness, scattering and finite time blowup. AIMS Mathematics, 2024, 9(11): 30230-30262. https://doi.org/10.3934/math.20241460

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Received: 22 September 2024
Revised: 10 October 2024
Accepted: 14 October 2024
Published: 24 October 2024
©2024 the Author(s), licensee AIMS Press.

This is an open access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0)