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Applied Mathematics & Optimization

, Volume 78, Issue 2, pp 219–265 | Cite as

Exponential Asymptotic Stability for the Klein Gordon Equation on Non-compact Riemannian Manifolds

  • C. A. Bortot
  • M. M. Cavalcanti
  • V. N. Domingos Cavalcanti
  • P. Piccione
Article
  • 180 Downloads

Abstract

The Klein Gordon equation subject to a nonlinear and locally distributed damping, posed in a complete and non compact n dimensional Riemannian manifold \((\mathcal {M}^n,\mathbf {g})\) without boundary is considered. Let us assume that the dissipative effects are effective in \((\mathcal {M}\backslash \Omega ) \cup (\Omega \backslash V)\), where \(\Omega \) is an arbitrary open bounded set with smooth boundary. In the present article we introduce a new class of non compact Riemannian manifolds, namely, manifolds which admit a smooth function f, such that the Hessian of f satisfies the pinching conditions (locally in \(\Omega \)), for those ones, there exist a finite number of disjoint open subsets \( V_k\) free of dissipative effects such that \(\bigcup _k V_k \subset V\) and for all \(\varepsilon >0\), \(meas(V)\ge meas(\Omega )-\varepsilon \), or, in other words, the dissipative effect inside \(\Omega \) possesses measure arbitrarily small. It is important to be mentioned that if the function f satisfies the pinching conditions everywhere, then it is not necessary to consider dissipative effects inside \(\Omega \).

Notes

Acknowledgements

Research of Marcelo M. Cavalcanti partially supported by the CNPq Grant 300631/2003-0. Research of Valéria N. Domingos Cavalcanti partially supported by the CNPq Grant 304895/2003-2. Research of Paolo Piccione partially supported by CNPq and Fapesp.

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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • C. A. Bortot
    • 1
  • M. M. Cavalcanti
    • 2
  • V. N. Domingos Cavalcanti
    • 2
  • P. Piccione
    • 3
  1. 1.Technological Centre of JoinvilleFederal University of Santa Catarina - Campuses JoinvilleJoinvilleBrazil
  2. 2.Department of MathematicsState University of MaringáMaringáBrazil
  3. 3.Department of MathematicsIME-Universidade de São PauloSão PauloBrazil

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