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

Saleh Almuthaybiri; Radhia Ghanmi; Tarek Saanouni; Saleh Almuthaybiri; Radhia Ghanmi; Tarek Saanouni

doi:10.3934/math.20241460

AIMS Mathematics

2024, Volume 9, Issue 11: 30230-30262. doi: 10.3934/math.20241460

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Research article

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

1.
Department of Mathematics, College of Science, Qassim University, Saudi Arabia
2.
University of Tunis El Manar, Faculty of Sciences of Tunis, 2092 Tunis, LR03ES04 Partial Differential Equations, Tunisia

Received: 22 September 2024 Revised: 10 October 2024 Accepted: 14 October 2024 Published: 24 October 2024
MSC : 35Q55

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.
- Schrödinger problem,
- energy-critical,
- nonlinear equations,
- fixed point method,
- global existence,
- scattering,
- blowup
Citation: Saleh Almuthaybiri, Radhia Ghanmi, Tarek Saanouni. On the nonlinear Schrödinger equation with critical source term: global well-posedness, scattering and finite time blowup[J]. AIMS Mathematics, 2024, 9(11): 30230-30262. doi: 10.3934/math.20241460

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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.

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