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

Metadata

DOI10.5802/ahl.60
Keywords damped wave equations, stabilization, semi-uniform decay, unique continuation property, small trapped sets, weak attractors

Abstract

We consider the semilinear damped wave equation

∂tt2u(x,t)+γ(x)∂tu(x,t)=Δu(x,t)-αu(x,t)-f(x,u(x,t)).

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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