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. BortotEmail author
  • M. M. Cavalcanti
  • V. N. Domingos Cavalcanti
  • P. Piccione


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 \(\Omega \).



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
    Email author
  • 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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