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Well-Posedness and Stability for Schrödinger Equations with Infinite Memory

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Abstract

We study in this paper the well-posedness and stability for two linear Schrödinger equations in d-dimensional open bounded domain under Dirichlet boundary conditions with an infinite memory. First, we establish the well-posedness in the sense of semigroup theory. Then, a decay estimate depending on the smoothness of initial data and the arbitrarily growth at infinity of the relaxation function is established for each equation with the help of multipliers method and some arguments devised in (Guesmia in J Math Anal Appl 382:748–760, 2011) and (Guesmia in Applicable Anal 94:184–217, 2015).

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Acknowledgements

This work was initiated during the visit in July–August 2017 of the third author to Concepción University, Chile, and finished during the visit of the fourth author in June 2018 to Lorraine—Metz university, France, and the visits in August 2018 of the third author to Concepción university, Chile, and Maringa University, Brazil. The authors thank Concepción, Lorraine—Metz and Maringa Universities for their kind support and hospitality.

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Authors and Affiliations

  1. Department of Mathematics, State University of Maringá, Maringá, PR, 87020-900, Brazil

    M. M. Cavalcanti & V. N. Domingos Cavalcanti

  2. Institut Elie Cartan de Lorraine, UMR 7502, Université de Lorraine, 3 Rue Augustin Fresnel, BP 45112, 57073, Metz Cedex 03, France

    A. Guesmia

  3. DIM and CI2MA, Universidad del Concepción, Concepción, Chile

    M. Sepúlveda

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  1. M. M. Cavalcanti
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Research of M. Sepúlveda was supported FONDECYT Grant No. 1220869, ECOS-ANID project C20E03, and by Centro de Modelamiento Matemático (CMM), ACE210010 and FB210005, BASAL funds for centers of excellence from ANID-Chile. V. N. Domingos Cavalcanti, M. M. Cavalcanti and M. Sepúlveda thank the support of MATH-AMSUD project 21-MATH-03 (CTMicrAAPDEs)

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Cavalcanti, M.M., Domingos Cavalcanti, V.N., Guesmia, A. et al. Well-Posedness and Stability for Schrödinger Equations with Infinite Memory. Appl Math Optim 85, 20 (2022). https://doi.org/10.1007/s00245-022-09864-1

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