In this paper, we consider the wave equation with zero order term. We use the compactness uniqueness argument and some result of I. Lasiecka and D. Tataru in  to prove, directly, the exponential decay rate of the perturbed energy.
Y. Guo and P. F. Yao, “Stabilization of Euler-Bernoulli Plate Equation with Variable Coefficients by Nonlinear Boundary Feedback,” Journal of Mathematical Analysis and Applications, Vol. 317, No. 1, 2006, pp. 50-70.