Optimal decay rate of the energy for wave equations with critical potential

We study the long time behavior of solutions of the wave equation with a variable damping term V(x)u(t) in the case of critical decay V(x) >= V-0(1 + vertical bar x vertical bar(2))(-1/2) (see condition (A) below). The solutions manifest a new threshold effect with respect to the size of the coefficient V-0: for 1 < V-0 < N the energy decay rate is exactly t(-V0), while for V-0 >= N the energy decay rate coincides with the decay rate of the corresponding parabolic problem.