Decay of semilinear damped wave equations: cases without geometric control condition
Annales Henri Lebesgue, Volume 3 (2020), pp. 1241-1289.


Keywords damped wave equations, stabilization, semi-uniform decay, unique continuation property, small trapped sets, weak attractors


We consider the semilinear damped wave equation


In this article, we obtain the first results concerning the stabilization of this semilinear equation in cases where γ does not satisfy the geometric control condition. When some of the geodesic rays are trapped, the stabilization of the linear semigroup is semi-uniform in the sense that e At A -1 h(t) for some function h with h(t)0 when t+. We provide general tools to deal with the semilinear stabilization problem in the case where h(t) has a sufficiently fast decay.


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