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 $K_{\tau }=(I-\tau \Delta {)}^{-1}(I-\Delta {)}^{-1}$. 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 $M_{\ast }=64{\pi }^2\tau /\chi$, while the sharp threshold theorem is formulated as an Adams/logarithmic-HLS 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.