Appl Math Optim DOI 10.1007/s00245-017-9405-5 Exponential Asymptotic Stability for the Klein Gordon Equation on Non-compact Riemannian Manifolds C. A. Bortot 1 · M. M. Cavalcanti 2 · V. N. Domingos Cavalcanti 2 · P. Piccione 3 © Springer Science+Business Media New York 2017 Abstract The Klein Gordon equation subject to a nonlinear and locally distributed damping, posed in a complete and non compact n dimensional Riemannian manifold (M n , g) without boundary is considered. Let us assume that the dissipative effects are effective in (M\) ∪ (\V ), where  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 ), for those ones, there exist a finite number of disjoint open subsets V k free of dissipative effects such that  k V k ⊂ V and for all ε> 0, meas (V ) ≥ meas () − ε, or, in other words, the dissipative effect inside  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 . 1 Introduction This paper addresses the well-posedness as well as sharp uniform decay rate estimates of the energy related to the Klein Gordon equation subject to a nonlinear and locally distributed damping, posed in a complete and non compact n dimensional Riemannian manifold (M n , g) without boundary: B C. A. Bortot c.bortot@ufsc.br 1 Technological Centre of Joinville, Federal University of Santa Catarina - Campuses Joinville, Joinville, SC 89218-035, Brazil 2 Department of Mathematics, State University of Maringá, Maringá, PR 87020-900, Brazil 3 Department of Mathematics, IME-Universidade de São Paulo, São Paulo, SP 05508-090, Brazil 123