Solution-independent oscillation time for a quarter-square membrane

Determine whether there exists an oscillation time independent of the solution for the vibrating membrane equation on the square $(0,\pi)\times(0,\pi)$ when observations are restricted to the sub-square $(0,\frac{\pi}{2})\times(0,\frac{\pi}{2})$; that is, determine whether one finite interval length guarantees that every nonzero solution takes both positive and negative values in that observation region.

Background

The general theorem in the paper guarantees that each individual nonzero membrane solution has a finite oscillation time on every nonempty open subset. For the specified quarter-square observation region, however, the authors distinguish this solution-dependent conclusion from the stronger assertion of a uniform bound valid for all solutions. They identify the latter as a problem raised in 1985 and state that it remains unresolved.

References

In the special case $\Omega = (0,\pi)\times (0,\pi)$, and $\omega = (0, \frac{\pi}{2})\times (0, \frac{\pi}{2}) $, we know by {\rm Theorem \ref{gen} that any solution $u$ of memb has a uniform oscillation time on $\omega$, but it is still unknown whether there is an oscillation time independent of $u$, a problem raised in 1985 and already mentioned in {\rm .

— Uniform oscillation properties of almost periodic functions  (2610.06515 - Haraux, 5 Oct 2026) in Final Remark, Section 7, “Back to vibrating membranes”