On the Schrödinger-Debye system in compact Riemannian manifolds

Marcelo Nogueira; Mahendra Panthee

doi:10.3934/cpaa.2020022

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2020, Volume 19, Issue 1: 425-453. Doi: 10.3934/cpaa.2020022

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On the Schrödinger-Debye system in compact Riemannian manifolds

Marcelo Nogueira and
Mahendra Panthee^,

Department of Mathematics, State University of Campinas, Campinas-SP, 13083-859, Brazil

^* Corresponding author
^* Corresponding author

Received: December 2018

Revised: May 2019

Published: July 2019

The first author is supported by CAPES and CNPq, Brazil. The second author is partially supported by FAPESP (2016/25864-6) Brazil and CNPq (308131/2017-7) Brazil

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Abstract

We consider the initial value problem (IVP) associated with the Schrödinger-Debye system posed on a $d$-dimensional compact Riemannian manifold $M $ and prove the local well-posedness result for given data $ (u_0, v_0)\in H^s(M)\times (H^s(M)\cap L^{\infty}(M))$ whenever $s>\frac{d}2-\frac12 $, $d\geq 2 $. For $d=2 $, we apply a sharp version of the Gagliardo-Nirenberg inequality in compact manifold to derive an a priori estimate for the $H^1 $-solution and use it to prove the global well-posedness result in this space.

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Mathematics Subject Classification: Primary: 35Q55; Secondary: 53C35.

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