Exponential Decay for a Time-Varying Coefficients Wave Equation with Dynamic Boundary Conditions

We consider a wave equation with variable coefficients in time and space in a bounded domain \(\Omega \) $\mathrm{\Omega }$ which has the smooth boundary \(\Gamma =\Gamma _0\cup \Gamma _1\) $\mathrm{\Gamma }={\mathrm{\Gamma }}_0\cup {\mathrm{\Gamma }}_1$ such that \({\overline{\Gamma }}_0\cap {\overline{\Gamma }}_1\ne \emptyset \) ${\overline{\mathrm{\Gamma }}}_0\cap {\overline{\mathrm{\Gamma }}}_1\ne \mathrm{\varnothing }$ . We study this system that has a homogeneous Dirichlet boundary on \(\Gamma _0\) ${\mathrm{\Gamma }}_0$ and a dynamic boundary on \(\Gamma _1\) ${\mathrm{\Gamma }}_1$ . The innovation of the paper lies in the coefficients which depends on the time variable and the singularities generated by changing the boundary conditions along the interface, thus we need some special techniques to deal with these difficulties. Under some geometric assumptions, the exponential decay result of the system is established by the Riemannian geometry method and the energy perturbation method.