Decay of semilinear damped wave equations:cases without geometric control condition

We consider the semilinear damped wave equation $\partial_{tt}^2 u(x,t)+\gamma(x)\partial_t u(x,t)=\Delta u(x,t)-\alpha u(x,t)-f(x,u(x,t))$. In this article, we obtain the first results concerning the stabilization of this semilinear equation in cases where $\gamma$ 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 $\mathrm{\parallel }{e}^{At}{A}^{-1}\mathrm{\parallel }\le h(t)$ for some function $h$ with $h(t)\to 0$ when $t\to +\mathrm{\infty }$. We provide general tools to deal with the semilinear stabilization problem in the case where $h(t)$ has a sufficiently fast decay.