On coupled non-linear Schrödinger systems with singular source term

Saleh Almuthaybiri; Tarek Saanouni; Saleh Almuthaybiri; Tarek Saanouni

doi:10.3934/math.20241353

AIMS Mathematics

2024, Volume 9, Issue 10: 27871-27895. doi: 10.3934/math.20241353

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Research article

On coupled non-linear Schrödinger systems with singular source term

Saleh Almuthaybiri ,
Tarek Saanouni ^,

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

Received: 24 August 2024 Revised: 13 September 2024 Accepted: 18 September 2024 Published: 26 September 2024
MSC : 35Q55

This work studies a coupled non-linear Schrödinger system with a singular source term. First, we investigate the question of the local existence of solutions. Second, one proves the existence of global solutions which scatter in some Sobolev spaces. Finally, one establishes the existence of non-global solutions. The main difficulty here is to overcome the regularity problem in the non-linearity. Indeed, because of the singularity of the source term, the classical contraction method in the energy space fails in such a regime. So, this paper is to fill such a gap in the literature. The argument follows ideas in T. Cazenave and I. Naumkin (Comm. Contemp. Math., 19 (2017), 1650038). This consists to remark that the singularity problem is only near the origin. So, one needs to impose that the solution stays away from zero. This is not trivial, since there is no maximum principle for the Schrödinger equation. The existence of global solutions which scatter follows with the pseudo-conformal transformation via the existence of local solutions. Finally, the existence of non-global solutions follows with the classical variance method.
- Schrödinger system,
- nonlinear equations,
- fixed point method,
- existence,
- uniqueness,
- scattering,
- approximation,
- blow-up
Citation: Saleh Almuthaybiri, Tarek Saanouni. On coupled non-linear Schrödinger systems with singular source term[J]. AIMS Mathematics, 2024, 9(10): 27871-27895. doi: 10.3934/math.20241353

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Abstract

This work studies a coupled non-linear Schrödinger system with a singular source term. First, we investigate the question of the local existence of solutions. Second, one proves the existence of global solutions which scatter in some Sobolev spaces. Finally, one establishes the existence of non-global solutions. The main difficulty here is to overcome the regularity problem in the non-linearity. Indeed, because of the singularity of the source term, the classical contraction method in the energy space fails in such a regime. So, this paper is to fill such a gap in the literature. The argument follows ideas in T. Cazenave and I. Naumkin (Comm. Contemp. Math., 19 (2017), 1650038). This consists to remark that the singularity problem is only near the origin. So, one needs to impose that the solution stays away from zero. This is not trivial, since there is no maximum principle for the Schrödinger equation. The existence of global solutions which scatter follows with the pseudo-conformal transformation via the existence of local solutions. Finally, the existence of non-global solutions follows with the classical variance method.

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