A sufficient condition for absence of mass quantization in a chemotaxis system with local sensing

TL;DR

This paper investigates conditions under which solutions to a chemotaxis system with local sensing blow up in infinite time, showing that mass quantization may not occur and providing insights into singularity formation.

Contribution

It introduces a sufficient condition for the absence of mass quantization in a chemotaxis system, contrasting with the Keller--Segel model, and analyzes infinite-time blowup and singularity behavior.

Findings

Unbounded      blowup solutions.

Mass concentration at the origin with potential mass exceeding 8.

Mass quantization may not occur in this system, unlike in Keller--Segel.

Abstract

We analyze blowup solutions in infinite time of the Neumann boundary value problem for the fully parabolic chemotaxis system with local sensing: \begin{equation*} \begin{cases} u_t = \Delta(e^{-v}u)\qquad &\mathrm{in}\ \Omega \times (0,\infty), v_t = \Delta v -v + u\qquad &\mathrm{in}\ \Omega \times (0,\infty), \end{cases} \end{equation*} where $\Omega$ is a ball in two-dimensional space and with nonnegative radially symmetric initial data. In the case of the Keller--Segel system which has a similar mathematical structure with our system, it was shown that the solutions blow up in finite time if and only if $L\log L$ for the first component $u$ diverges in finite time. On the other hand, focusing on the variational structure induced by a signal-dependent motility function $e^{-v}$, we show that an unboundedness of $\int_\Omega e^v dx$ for the second component $v$ gives rise to blowup solutions in infinite time under the assumption of radial symmetry. Moreover we prove mass concentration phenomena at the origin. It is shown that the radially symmetric solutions of our system develop a singularity like a Dirac delta function in infinite time. Here we investigate the weight of this singularity. Consequently it is shown that mass quantization may not occur; that is, the weight of the singularity can exceed $8\pi$ under the assumption of a uniform-in-time lower bound for the Lyapunov functional. This type of behavior cannot be observed in the Keller--Segel system.