Long time dynamics of defocusing energy critical 3 + 1 dimensional wave equation with potential in the radial case

TL;DR

This paper studies the long-term behavior of radial solutions to the defocusing energy-critical wave equation with potential in 3+1 dimensions, showing convergence to steady states using energy inequalities.

Contribution

It extends the analysis of wave equations by proving convergence to steady states in the presence of potentials, including the finiteness of steady states for generic potentials.

Findings

Solutions converge to steady states modulo free radiation.

Finitely many steady states exist for generic potentials.

Solutions tend to a single steady state over time.

Abstract

Using 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.