Existence and asymptotic stability of a generic Lotka-Volterra system with nonlinear spatially heterogeneous cross-diffusion

The paper rigorously proves global bounded classical solutions and a complete stability classification for the three-compartment prey–taxis system with spatially heterogeneous predation and taxis in any dimension, using weighted Lpε estimates, heat-kernel bounds, and a generalized LaSalle principle. Its main results (Theorems 1.1–1.2) and the key technical lemmas are internally consistent and carefully justified, including the delicate handling of variable-coefficient cross-diffusion terms ∇(χi(x)h(·)∇·) and their derivatives. By contrast, the candidate solution’s Phase 1 relies on off-the-shelf “triangular cross-diffusion” W1,p-bounds and a Moser–Alikakos iteration without addressing the crucial variable-coefficient terms (∇χi, h′) or the need for weighted-in-time norms. It also incorrectly treats taxis fluxes as if χi were constant when invoking semigroup bootstraps. While its Phase 2 stability conclusions align in spirit with the paper, the candidate’s global existence and uniform-boundedness argument skips the core difficulties resolved in the paper and hence is incomplete in this general setting.