Inhomogeneous NLS with partial harmonic confinement

Saleh Almuthaybiri; Tarek Saanouni; Saleh Almuthaybiri; Tarek Saanouni

doi:10.3934/math.2025450

AIMS Mathematics

2025, Volume 10, Issue 4: 9832-9851. doi: 10.3934/math.2025450

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Research article

Inhomogeneous NLS with partial harmonic confinement

Saleh Almuthaybiri ,
Tarek Saanouni ^,

Department of Mathematics, College of Science, Qassim University, Saudi Arabia

Received: 19 February 2025 Revised: 11 April 2025 Accepted: 23 April 2025 Published: 27 April 2025
MSC : 35Q55

We investigate the inhomogeneous nonlinear Schrödinger equation with partial harmonic confinement. First, we present a global well-posedness result for small data in the intercritical regime. Second, we obtain a threshold of global existence versus finite-time blow-up in the mass-critical regime. Finally, we prove the $ L^2 $ concentration of the mass-critical non-global solution with minimal mass. The challenge is to address the fact that the standard scale invariance is broken by the partial confinement. We use the associated ground state without potential in order to describe the threshold of global versus non-global existence of solutions.
- Schrödinger equation,
- partial harmonic confinement,
- global existence,
- blow-up,
- ground state,
- nonlinear equations,
- approximation
Citation: Saleh Almuthaybiri, Tarek Saanouni. Inhomogeneous NLS with partial harmonic confinement[J]. AIMS Mathematics, 2025, 10(4): 9832-9851. doi: 10.3934/math.2025450

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Abstract

We investigate the inhomogeneous nonlinear Schrödinger equation with partial harmonic confinement. First, we present a global well-posedness result for small data in the intercritical regime. Second, we obtain a threshold of global existence versus finite-time blow-up in the mass-critical regime. Finally, we prove the $ L^2 $ concentration of the mass-critical non-global solution with minimal mass. The challenge is to address the fact that the standard scale invariance is broken by the partial confinement. We use the associated ground state without potential in order to describe the threshold of global versus non-global existence of solutions.

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