Long Time Dynamics of Defocusing Energy Critical 3 + 1 Dimensional Wave Equation with Potential in the Radial Case

Using the channel of energy inequalities developed by T. Duyckaerts, C. Kenig and F. Merle, we prove that, modulo a free radiation, any finite energy radial solution to the defocusing energy critical wave equation with radial potential in 3 + 1 dimensions converges to the set of steady states as time goes to infinity. For generic potentials, we prove there are only finitely many steady states, and in this case, modulo some free radiation, the solution converges to one steady state as time goes to infinity.