Elliptic criticality versus Volterra memory in indirect chemotaxis cascades

Abstract

Indirect signal production is often treated as a higher-order variant of classical Keller–Segel chemotaxis, but its critical structure depends strongly on how the signal cascade is closed. This paper separates two asymptotic regimes of a two-stage signalling mechanism. In the parabolic–elliptic–elliptic limit, the chemoattractant is generated by the self-adjoint fourth-order operator K _τ = ( I − τ Δ) ⁻¹ ( I − Δ) ⁻¹ . We prove its spectral positivity, entropy-dissipation structure, fourth-order principal scaling, and logarithmic kernel singularity in four dimensions. Consequently, the correct critical space is L ^{N /4} , and N = 4 is the mass-critical dimension. A concentration calculation identifies the natural threshold candidate M _∗ = 64 π ² τ / χ , while the sharp threshold theorem is formulated as an Adams/logarithmic-Hardy–Littlewood–Sobolev open problem. In contrast, the mixed elliptic–parabolic cascade cannot be reduced to a static fourth-order kernel. Its eliminated signal is a Volterra memory operator whose near-diagonal multiplier has the same order as the classical Keller–Segel drift. Thus its critical theory must be based on mixed space–time estimates, not static elliptic scaling. Numerical experiments support our operator-level distinction.