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

 ${\partial }_{tt}^{2}u\left(x,t\right)+\gamma \left(x\right){\partial }_{t}u\left(x,t\right)=\Delta u\left(x,t\right)-\alpha u\left(x,t\right)-f\left(x,u\left(x,t\right)\right)\phantom{\rule{0.166667em}{0ex}}.$

In this article, we obtain the first results concerning the stabilization of this semilinear equation in cases where $\gamma$ 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 $\parallel {e}^{At}{A}^{-1}\parallel \le h\left(t\right)$ for some function $h$ with $h\left(t\right)\to 0$ when $t\to +\infty$. We provide general tools to deal with the semilinear stabilization problem in the case where $h\left(t\right)$ has a sufficiently fast decay.

On considère l’équation des ondes amorties

 ${\partial }_{tt}^{2}u\left(x,t\right)+\gamma \left(x\right){\partial }_{t}u\left(x,t\right)=\Delta u\left(x,t\right)-\alpha u\left(x,t\right)-f\left(x,u\left(x,t\right)\right)\phantom{\rule{0.166667em}{0ex}}.$

Dans cet article, on obtient les premiers résultats concernant la stabilisation de cette équation dans des cas où $\gamma$ 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 $\parallel {e}^{At}{A}^{-1}\parallel \le h\left(t\right)$ pour une certaine fonction $h$ avec $h\left(t\right)\to 0$ quand $t\to +\infty$. On donne une méthode générale pour traiter le problème de la stabilisation semilinéaire dans le cas où $h\left(t\right)$ décroit suffisamment.

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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