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Asymptotic behavior of cubic defocusing Schrödinger equations on compact surfaces

  • Marcelo M. Cavalcanti
  • Wellington J. Corrêa
  • Valéria N. Domingos Cavalcanti
  • Maria R. Astudillo Rojas
Article

Abstract

We are concerned with the asymptotic behavior of two different cubic, defocusing and damped nonlinear Schrödinger equations on compact Riemannian manifolds without boundary. Two mechanisms of locally distributed damping are considered: a weak damping and a stronger one. In the first problem, we consider a two-dimensional case and prove that the corresponding energy functional goes to zero as time goes to infinity. The proof is based on a result of propagation of singularities due to Dehman et al. (Math Z 254(4):729–749, 2006) and Strichartz type inequalities due to Burq et al. (Am J Math 126(3):569–605, 2004), combined with new ingredients which come from the observability inequality associated with the linear problem. When a stronger damping is in place, we show that the energy functional decays exponentially to zero and for this purpose a forced smoothing effect due to Aloui (Collect Math 59(1):53–62, 2008) plays an essential role in the proof.

Mathematics Subject Classification

35B35 35Q55 

Notes

Acknowledgements

The authors are grateful to the anonymous referees for their suggestions and fruitful comments.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Marcelo M. Cavalcanti
    • 1
  • Wellington J. Corrêa
    • 2
  • Valéria N. Domingos Cavalcanti
    • 1
  • Maria R. Astudillo Rojas
    • 1
  1. 1.Department of MathematicsState University of MaringáMaringáBrazil
  2. 2.Academic Department of MathematicsFederal Technological University of ParanáCampo MourãoBrazil

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