Zero-solution uniqueness for the backward heat equation with homogeneous initial and boundary conditions

Prove that the only solution W ≡ 0 exists for the initial–boundary value problem W_t + ΔW = 0 in Ω × [0, T] with W(·,0) = 0 and with one of the homogeneous boundary conditions on ∂Ω: (i) Dirichlet W|_{∂Ω} = 0, or (ii) Neumann ∂W/∂n|_{∂Ω} = 0, or (iii) Robin ∂W/∂n + σW|_{∂Ω} = 0 for σ > 0. Establishing this would justify the step needed in the Neumann-boundary analysis linked to equation (3.54).

Background

In Section 3, while treating boundary integral equations for the Navier–Stokes reformulation, the analysis of the Neumann problem leads to auxiliary kernels satisfying a backward parabolic equation h'_t + Δh' = 0 rather than the forward heat equation. To carry over the argument used in the Dirichlet case, the authors would need the uniqueness of the trivial solution for a corresponding homogeneous initial–boundary value problem.

They note that their method would work if one could assert that the only solution to the backward heat equation with zero initial data and homogeneous Dirichlet/Neumann/Robin boundary conditions is the zero function, but they explicitly state uncertainty about this step. They subsequently circumvent the issue via a time-reversal transform, but do not resolve the uniqueness question for the original formulation (3.54).