A frequency approach for stabilization of one-dimensional degenerate wave equation

In this paper, we are concerned with the study of stabilization problem for the following strongly degenerate wave equation in one space dimension $$w_{tt}(x,t)-{(x^{\alpha }{w}_x(x,t))}_x=0$$ where ${\bf\alpha\in [1,2)}$. Thus, using a frequency domain method inspired from \cite{BT}, we prove the polynomial decays of its total energy with $t^{-\text{nicefrac}12}$ decay rate.