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

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.

On considère l’équation des ondes amorties

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

Dans cet article, on obtient les premiers résultats concernant la stabilisation de cette équation dans des cas où γ ne vérifie pas la condition de contrôle géométrique. Quand certains des rayons géodésiques sont captés, la stabilisation du semigroupe est semi-uniforme dans le sens où e At A -1 h(t) pour une certaine fonction h avec h(t)0 quand t+. On donne une méthode générale pour traiter le problème de la stabilisation semilinéaire dans le cas où h(t) décroit suffisamment.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/ahl.60
Classification: 53C28,  53C26,  32Q45
Keywords: damped wave equations, stabilization, semi-uniform decay, unique continuation property, small trapped sets, weak attractors
Joly, Romain 1; Laurent, Camille 2, 3

1 Université Grenoble Alpes, CNRS, Institut Fourier, F-38000 Grenoble, (France)
2 CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France
3 UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France
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Joly, Romain; Laurent, Camille. Decay of semilinear damped wave equations: cases without geometric control condition. Annales Henri Lebesgue, Volume 3 (2020), pp. 1241-1289. doi : 10.5802/ahl.60. http://www.numdam.org/articles/10.5802/ahl.60/

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