Parabolic analogue of elliptic growth condition HP3

Determine whether a parabolic counterpart exists for the elliptic potential-growth assumption HP3 in Liouville theorems for the Finsler $(p,q)$-Laplacian parabolic inequality.

Background

The paper establishes elliptic Liouville theorems under three potential-growth hypotheses, HP1, HP2, and HP3. For the associated parabolic inequality, it introduces only HP4 and HP5, which correspond respectively to the elliptic assumptions HP1 and HP2.

The authors explicitly note that HP3 has no known parabolic counterpart. Thus, an unresolved problem is to determine whether an analogous parabolic growth condition—particularly one allowing the type of stronger-than-critical logarithmic growth compensated by an exponential factor in HP3—can be formulated and used to obtain a corresponding Liouville theorem.

References

As mentioned in , HP4 and HP5 are the counterparts of HP1 and HP2 above, respectively; while there is no known parabolic counterpart of the elliptic assumption HP3.

— On Liouville problems for $(p,q)$-Laplacian inequalities on Finsler measure spaces  (2609.04780 - Wang et al., 4 Sep 2026) in Remark following the definitions of HP4 and HP5, Section 1, subsection “Our main results”