Decay estimates for a degenerate wave equation with a dynamic nonlocal control acting on the degenerate boundary

Abstract

In this paper, we consider a one-dimensional weakly degenerate wave equation with a  dynamic nonlocal boundary control
of fractional type acting at a degenerate point. First We show well-posedness by using the semigroup theory.
Next, we show that our system is not uniformly stable by spectral analysis.
Hence, we look for a polynomial decay rate for smooth initial data by using a result due Borichev and Tomilov
which reduces the problem of estimating the rate of energy decay to finding a growth bound for the resolvent of the
generator associated with the semigroup. This analysis proves that the degeneracy affect the energy decay rates.