Asymptotic persistence of Strichartz estimates at the critical quintic level

Determine whether Strichartz estimates for three-dimensional critical quintic wave equations remain valid uniformly over arbitrarily long time intervals, in contrast to their established local-in-time use.

Background

The paper explains that, at the critical quintic level, local Strichartz estimates depend on the particular solution rather than solely on its energy norm, so they do not immediately yield uniform long-time control. The unresolved issue concerns whether the dispersive space-time estimates can persist asymptotically in time for critical quintic wave dynamics.

The authors then distinguish the subquintic regime, treated in the paper, where long-time control of the Strichartz norms can be obtained. Thus, the stated open problem concerns the unresolved critical-level long-time theory rather than the subquintic results established here.

References

At the critical quintic level, these estimates do not provide any control of long-time behavior-as the estimates themselves depend on the solution rather than the norm Section 3. Thus, the question of whether Strichartz estimates survive asymptotically in time is still an open problem [in contrast to energy estimates].

— Global Well-Posedness and Asymptotic Behavior of Energy-Damped Subquintic Wave Equations  (2609.29778 - Lasiecka et al., 24 Sep 2026) in Section 1, subsection “General comments,” subsection “Quintic and subquintic forcings”