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

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 $$\Omega $$ .