Exponential decay under the general drift hypothesis (Hy)

Establish whether exponential decay estimates analogous to the $L^2$ estimate in Theorem 1.1(ii) hold for linear parabolic initial-boundary value problems under the more general drift condition (Hy), without the structural condition (S).

Background

The paper proves exponential L2L^2-decay under the structural condition (S), which permits the construction of a positive weight and a divergence-free transformation for the drift. Earlier in the paper, existence and uniqueness are established under the broader hypothesis (Hy), but the resulting estimates have exponentially growing rather than decaying bounds.

The unresolved issue is whether the stronger decay conclusion can be extended from (S) to (Hy). The authors identify the structural condition (S) as essential to their current weighted-semigroup method, so resolving this problem would require either extending that construction or developing a different approach.

References

Achieving this decay behavior in our framework is critically dependent on the drift coefficient $\mathbf{H}$ satisfying the structural condition {\bf (S)}. Whether exponential decay estimates similar to expdecaacer can be obtained under the more general condition {\bf (Hy)} remains an interesting open problem.

— A weighted semigroup approach to exponential stability in linear parabolic equations  (2609.05173 - Lee, 4 Sep 2026) in Remark following Theorem 2.4, Section 2 (before Section 3)

While our results demonstrate the robustness of exponential decay even in the presence of nonsymmetric drifts, whether such drifts quantitatively accelerate convergence remains an open question.

— A weighted semigroup approach to exponential stability in linear parabolic equations  (2609.05173 - Lee, 4 Sep 2026) in Section Discussion