Shock-type singularity of the hyperbolic-parabolic chemotaxis system

2025/01/16 by Woojae Lee, Lee, Woojae
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Mathematical Biology Tumor Growth

paper · pdf · doi:10.48550/arxiv.2501.09656

openalex publication_date 2025/01/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper deals with the hyperbolic-parabolic chemotaxis (HPC) model, which is a hydrodynamic model describing vascular network formation at the early stage of the vasculature. We study analytically the singularity formation associated with the shock-type structure, which was numerically observed by Filbet, Laurençot, and Perthame \citefilbet2005derivation and Filbet and Shu \citefilbet2005approximation. We construct the blow-up profile in a 1D HPC system on ℝ as follows: The blow-up profile is stable in the sense of H^m topology (m≥ 5) prior to the occurrence of the singularity. For the first singularity, while the density and velocity (ρ, u) of endothelial cells themselves remain bounded, the gradients of the density and velocity blow up. The chemoattractant concentration ϕ has C² regularity. However, the density and velocity with C^ (1)/(3) regularity exhibit a cusp singularity at a unique blow-up point, the location and time of which are explicitly estimated. Furthermore, the HPC system is C¹ differentiable except in any neighborhood of the blow-up point.

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