Stability for wave equation with localized damping revisited

Resumo: In this talk we revisit some classical problems involving the wave equation to show another way to prove the stability of the problems. We start considering the n-dimensional linear wave equation in a bounded domain subject to a locally distributed linear damping term. In this case, we proved, via semigroup results (Gearhart Theorem), that the energy decays exponentially. For our surprise, this result has never been proved by this methodology so far. After this, using the linear case, we proved the stability to the wave equation now subject to a locally distributed nonlinear damping term. The second case considered is when the domain is whole space $\mathbb{R}^N$. In this situation, we proved two results. First, using semigroups results with full damping out of a compact set. The second case (when the domain is whole space) the goal here is that we removed damping. Precisely, given a positive real number $M>0$, we show that for any compact set $K$ of $\mathbb{R}^N$, it is possible to build a region $\Xi$ free of damping, with $meas(\Xi)=M$ (measure of $\Xi$), such that $\Xi$ is globally distributed. Finally, we also considered a case with an unbounded domain with finite measure. In both cases where the undamped region is unbounded it possesses a finite measure we considered microlocal analysis tools combined with Egorov's theorem.