A sharp criterion and complete classification of global-in-time solutions and finite time blow-up of solutions to a chemotaxis system in supercritical dimensions

TL;DR

The paper analyzes a chemotaxis system with indirect signal production in $\mathbb{R}^d$ for supercritical dimensions, proving a sharp threshold in the scaling-critical Morrey norms that separates global existence from finite-time blow-up. Because the system remains genuinely parabolic-parabolic, the authors develop radial comparison principles and mass-function techniques, linking the dynamics to a singular stationary state and to the fourth-order biharmonic problem $\Delta^2 \phi = e^{\phi}$. They establish that the Morrey norms of $e^{\phi}$ and $(-\Delta)\phi$ serve as critical thresholds, yielding a complete classification of long-time behavior in dimensions $d\ge5$, with dimension-dependent refinements (notably sharp thresholds in $d\ge13$ and smallness regimes for $5\le d\le12$). The analysis integrates stationary-state structure, monotone mass-function dynamics, and radial symmetry to derive global existence results, finite-time blow-up, and convergence to stationary states, contributing new tools for essentially parabolic-parabolic chemotaxis systems. The results have significant implications for understanding pattern formation and aggregation in high-dimensional chemotaxis models with indirect signaling.

Abstract

We consider the chemotaxis system with indirect signal production in the whole space, \begin{equation}\label{abst:p}\tag{$\star$} \begin{cases} u_t = Δu - \nabla \cdot (u\nabla v),\\ 0 = Δv + w,\\ w_t = Δw + u \end{cases} \end{equation} with emphasis on supercritical dimensions. In contrast to the classical parabolic-elliptic Keller--Segel system, where the analysis can be reduced to a single equation, the above system is essentially parabolic-parabolic and does not admit such a reduction. In this paper, we establish a sharp threshold phenomenon separating global-in-time existence from finite time blow-up in terms of scaling-critical Morrey norms of the initial data. In particular, we prove the existence of singular stationary solutions and show that their Morrey norm values serve as the critical thresholds determining the long-time behavior of solutions. Consequently, we identify new critical exponents at which the long-time behavior of solutions changes. This yields a complete classification of the long-time behavior of solutions, providing the first such results for the essentially parabolic-parabolic chemotaxis system \eqref{abst:p} in supercritical dimensions.

Theorems & Definitions (66)

• Theorem 1.1: Singular stationary solutions
• Remark 1.2
• Theorem 1.3
• Remark 1.4
• Theorem 1.5
• Remark 1.6
• Remark 1.7
• Theorem 1.8
• Remark 1.9
• Theorem 1.10