Soliton resolution for the energy critical damped wave equations in the radial case

We consider energy-critical damped wave equation \begin{equation*} \partial_{tt}u-\Delta u+\alpha \partial_t u=\left|u\right|^{\frac{4}{D-2}}u \end{equation*} with radial initial data in dimensions $D\geq 4$.

The equation has a nontrivial radial stationary solution $W$, called the ground state, which is unique up to sign and scale.

We prove that any bounded energy norm solution behaves asymptotically as a superposition of the modulated ground states and a radiation term.

In the global case, particularly, the solution converges to a pure multi-bubble due to the damping effect.