Backward uniqueness for the fast diffusion equation (1.8)

Establish backward uniqueness for solutions of the fast diffusion equation ∂_t u = (1/m) Δ(u^m) in ℝ^n with exponent 0 < m < 1 (equivalently α = (m−1)/2 ∈ (−1/2,0)), under appropriate conditions on the initial data, to enable a PDE-based derivation of equality conditions for the Borell–Brascamp–Lieb inequality beyond the heat equation case.

Background

In developing a PDE approach to equality conditions, the authors successfully leverage eventual log-concavity and backward uniqueness for the heat equation (α=0). They aim to extend the method to fast diffusion (α∈[−1/n,0)), noting that eventual power concavity results exist.

However, they explicitly state that the backward uniqueness counterpart for the fast diffusion equation is, to their knowledge, not established, which blocks a rigorous completion of the equality-case argument for α<0.