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

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

### Abstract

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.

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