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

Abstract

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) ≥ V₀(1 + |x|²)^−1/2 (see condition (A) below). The solutions manifest a new threshold effect with respect to the size of the coefficient V₀: for 1 < V₀ < N the energy decay rate is exactly t^−V₀, while for V₀ ≥ N the energy decay rate coincides with the decay rate of the corresponding parabolic problem.