Local well-posedness for the inhomogeneous nonlinear Schrödinger equation

Lassaad Aloui; Slim Tayachi

doi:10.3934/dcds.2021082

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2021, Volume 41, Issue 11: 5409-5437. Doi: 10.3934/dcds.2021082

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Local well-posedness for the inhomogeneous nonlinear Schrödinger equation

Lassaad Aloui^1, and
Slim Tayachi^2, ,

1.
Université de Tunis El Manar, Faculté des Sciences de Tunis, Département de Mathématiques, 2092 Tunis, Tunisie
2.
Université de Tunis El Manar, Faculté des Sciences de Tunis, Département de Mathématiques, Laboratoire Équations aux, Dérivées Partielles LR03ES04, 2092 Tunis, Tunisie

^* Corresponding author
^* Corresponding author

Received: October 2020

Revised: March 2021

Early access: May 2021

Published: November 2021

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Abstract

We consider the Cauchy problem for the inhomogeneous nonlinear Schrödinger equation $ i\partial_t u +\Delta u = \mu |x|^{-b}|u|^\alpha u,\; u(0)\in H^s({\mathbb R}^N),\; N\geq 1,\; \mu\in {\mathbb C},\; \; b>0 $ and $ \alpha>0. $ Only partial results are known for the local existence in the subcritical case $ \alpha<(4-2b)/(N-2s) $ and much more less in the critical case $ \alpha = (4-2b)/(N-2s). $ In this paper, we develop a local well-posedness theory for the both cases. In particular, we establish new results for the continuous dependence and for the unconditional uniqueness. Our approach provides simple proofs and allows us to obtain lower bounds of the blowup rate and of the life span. The Lorentz spaces and the Strichartz estimates play important roles in our argument. In particular this enables us to reach the critical case and to unify results for $ b = 0 $ and $ b>0. $

Keywords:

Inhomogeneous nonlinear Schrödinger equation,
local well-posedness,
continuous dependence,
unconditional uniqueness,
blow-up rate,
life span,
global existence.

Mathematics Subject Classification: Primary: 35Q55, 35B30; Secondary: 35Q70.

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