Energy decay for a wave equation of variable coefficients with logarithmic nonlinearity source term

We investigate the variable-coefficient wave equation with nonlinear acoustic boundary conditions and logarithmic nonlinearity source term. In addition, the damping is nonlinear, and plays a role only in part of the boundary. Using the semigroup theory, we state the existence and uniqueness of the local solutions. Accordingly, we prove that under some assumptions on the initial data, the local solution can be extended to be global. Particularly, it has been shown that the decay rates of the system are given implicitly as solutions to a first-order ODE by applying the Riemannian geometry method.