In this article, we prove the exponential stabilization of the semilinear wave equation with a damping effective in a zone satisfying the geometric control condition only. The nonlinearity is assumed to be subcritical, defocusing and analytic. The main novelty compared to previous results, is the proof of a unique continuation result in large time for some undamped equation. The idea is to use an asymptotic smoothing effect proved by Hale and Raugel in the context of dynamical systems. Then, once the analyticity in time is proved, we apply a unique continuation result with partial analyticity due to Robbiano, Zuily, Tataru and Hörmander. Some other consequences are also given for the controllability and the existence of a compact attractor

We study the decay of the semigroup generated by the damped wave equation in an unbounded domain. We first prove under the natural geometric control condition the exponential decay of the semigroup. Then we prove under a weaker condition the logarithmic decay of the solutions (assuming that the initial data are smoother). As corollaries, we obtain several extensions of previous results of stabilisation and control.Onétudie la décroissance du semi-groupe des ondes amorties dans un domaine non borné. Notre premier résultat est que, sous une hypothèse naturelle de contrôle géométrique, le semigroupe décroît exponentiellement vite. On démontre ensuite sous une hypothèse plus faible la décroissance logarithmique des solutions associéesà des données initiales plus régulières. On obtient en corollaire plusieurs généralisations de résultats de stabilisation et de contrôle.