Removing compact support in the PDE proof of the Prekopa–Leindler equality case

Ascertain whether the assumption of compact support on the initial data φ used in the PDE-based proof of the equality condition for the Prekopa–Leindler inequality (α = 0) can be dropped while retaining applicability of eventual log-concavity and backward uniqueness tools for the heat equation.

Background

To recover equality conditions for PL via PDE methods, the authors exploit eventual strong log-concavity of heat-flow solutions and backward uniqueness. Their present proof assumes the initial data are continuous with compact support to apply known eventual log-concavity results.

They explicitly note uncertainty about removing this compact support assumption, suggesting that suitable approximations preserving both the inequality premise and equality could work, but leaving the question unresolved.