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We study the decay of the global energy for the damped Klein-Gordon equationon non-compact manifolds with finitely many cylindrical and subconic ends up tobounded perturbation. We prove that under the Geometric Control Condition, thedecay is exponential, and that under the weaker Network Control Condition, thedecay is logarithmic, by developing the global Carleman estimate with multipleweights.

Exponential decay for damped Klein-Gordon equations on asymptotically cylindrical and conic manifolds