Parabolic unique continuation for the p-Laplacian flow

Determine whether a solution of the parabolic equation u_t=Δ_pu+f(u) on Ω×(0,T), with 1<p<2 and f satisfying the stated growth condition, that vanishes on a nonempty space-time cylinder ω×(t_1,t_2) must vanish throughout Ω at every time t∈(t_1,t_2).

Background

The elliptic equation studied in the paper is the stationary counterpart of the parabolic p-Laplacian flow u_t=Δ_pu+f(u). The paper proves elliptic unique continuation in the plane but does not resolve the analogous parabolic question for 1<p<2.

The unresolved issue is whether vanishing on a nonempty open space-time cylinder forces the entire spatial slice to vanish at each time in the interval. The authors explain that the answer is positive for p=2, whereas their elliptic method for 1<p<2 would require a strong local bound on u_t that is generally unavailable.

References

and for the parabolic flow the corresponding question is open: if $u$ vanishes on a nonempty open subset $\omega\times(t_1,t_2)$, does $u(\cdot,t)$ vanish in $\Omega$ for every $t\in(t_1,t_2)$?

— Unique continuation, nonexistence and bubbling for the critical $p$-Laplace equation in the plane  (2609.34204 - Mercuri, 28 Sep 2026) in Section 1, subsection “Related questions,” item 3