On the stabilization of some degenerate vibrating equation by fractional damping

Abstract

We study the stabilization of the degenerate wave equation $u_{tt}−(\sqrt{1−x^2})u_x)_x =0$ with $x ∈ (−1,1)$, by a fractional boundary damping acting at $x = 1$. Thus, using semigroup theory and method inspired from Rozendaal, stahn and Seifertr. We prove the logarithm decays of its total energy with (lnt)^{−2} decay rate where  $0<\alpha< 1$.