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| 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 $(\mathcal{M}\backslash \overline{\Omega}) \cup (\Omega \backslash V)$, 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 {\em 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 $\varepsilon>0$, $meas(V)\geq meas(\Omega)-\varepsilon$, 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 $\Omega$. |
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