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

This paper establishes a precise threshold criterion based on initial data norms that determines whether solutions to a chemotaxis system in supercritical dimensions exist globally or blow up in finite time, providing a complete classification.

Contribution

It introduces a sharp threshold criterion for global existence versus blow-up in a parabolic-parabolic chemotaxis system, extending understanding beyond classical models.

Findings

Identified critical Morrey norm thresholds for solution behavior.

Constructed singular stationary solutions serving as thresholds.

Provided a complete classification of long-time dynamics in supercritical dimensions.

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 = \Delta u - \nabla \cdot (u\nabla v),\\ 0 = \Delta v + w,\\ w_t = \Delta w + 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.