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 $V(x)\geq V_0(1+|x|^2)^{-1/2}$ (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 $t^{-V_0}$, while for $V_0\geq N$ the energy decay rate coincides with the decay rate of the corresponding parabolic problem.