| Contents |
| The Schr\"odinger equation subject to a nonlinear and locally
distributed damping, posed in a connected, complete and non
compact $n$ dimensional Riemannian manifold
$(M,\mathbf{g})$ is considered. Assuming that
$(M,\mathbf{g})$ is non-trapping and, in addition, that
the damping term is effective in $M \backslash \Omega$, where
$\Omega\subset \subset M$ is an open bounded and connected subset
with smooth boundary $\partial\Omega$, such that
$\overline{\Omega}$ is a compact set, exponential and uniform decay
rates of the $L^2-$level energy are established. The main
ingredients in the proof of the exponential stability are: (A)~ an
unique continuation property for the linear problem (as in Triggiani
and Xu (Contemporary Math./2007)); and (B)~ a local smoothing effect for
the linear and non-homogeneous associated problem (as in Burq, Gerard and Tzvetkov (AIHP/2004 )) |
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