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

## TL;DR

This paper investigates the stabilization of semilinear damped wave equations without the geometric control condition, providing new insights into cases with trapped geodesic rays and establishing conditions for semi-uniform stabilization.

## Contribution

It introduces the first results on semilinear stabilization without geometric control, especially when geodesic rays are trapped, and develops tools for fast decay scenarios.

## Key findings

- Linear semigroup stabilization is semi-uniform with trapped rays.
- Provided conditions for semilinear stabilization with fast decay functions.
- Established new methods for handling non-geometric control cases.

## Abstract

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 $\|e^{At}A^{-1}\|\leq h(t)$ for some function $h$ with $h(t)\rightarrow 0$ when $t\rightarrow +\infty$. We provide general tools to deal with the semilinear stabilization problem in the case where $h(t)$ has a sufficiently fast decay.