The food seeking behavior of slime mold: a macroscopic approach

The paper’s Theorem 2 asserts that for A,B<0 and V,W in H^2∩L∞ on a bounded connected domain, the only stationary state is 0 when C is sufficiently large, proved via a fixed-point/contraction on the Poisson map Δρ̃=G(ρ) (definition of G and the plan are clear in their write-up of 3.2 High diffusion stationary states and Theorem 2) . However, the contraction step relies on the claim ||G(ρ1)−G(ρ2)||_{L2} ≤ β ||G(ρ1)−G(ρ2)||_{L1} (their (33)), which reverses the standard inequality on bounded domains (one has ||f||_{L1} ≤ |Ω|^{1/2}||f||_{L2}, not the other way), so the argument as written does not close . In contrast, the candidate solution gives a correct energy argument under explicit, stronger hypotheses (ρ≥0, convex W so ΔW≥0, and d≤4 for H^1_0↪L^4), yielding an explicit threshold C0 and uniqueness of the zero state. Thus, the model’s proof is correct under its stated assumptions; the paper’s proof is incomplete as written.