Energy Decay Rate of the Wave Equations on Riemannian Manifolds with Critical Potential

Decay of the energy for the Cauchy problem of the wave equation on Riemannian manifolds with a variable damping term \(V(x)u_{t}\) is considered, where \(V(x) \ge V_{0}(1 + \rho ^{2})^{-\frac{1}{2}}\)(\(\rho \) being a distance function under the Riemannian metric). Some relations among the decay rates of energy, the size of the coefficients \(V_{0}\), and the radial curvatures of the Riemannian metric are presented.