Classification of self-similar singular solutions with large mass for Keller-Segel model with signal consumption

TL;DR

This paper investigates the existence, decay, and singularity properties of self-similar solutions in the Keller-Segel model with signal consumption, revealing how solutions behave with respect to mass, singularity, and regularity.

Contribution

It provides a detailed analysis of self-similar solutions with arbitrary mass, including their decay rates and singularity types, across different dimensions and parameter values.

Findings

$u$ behaves like a heat kernel with Dirac delta initial singularity

$u$ converges to zero in $L^p$ norm as time increases

$v$ exhibits varying regularity depending on $eta$ and dimension

Abstract

In this paper, we concentrate on investigating the self-similar singular solutions of Keller-Segel model with signal consumption ($-uv^{\alpha}$) and singular sensitivity. We perform a detailed exploration into the existence and decay rate of self-similar solutions, particularly, the permissibility of arbitrary mass for these solutions across all possible cases. Based on these findings, we can delve deeper into verifying that these self-similar solutions $(u, v)$ exhibit varying degrees of singularity depending on the value of $\alpha$ and the spatial dimension. Our analysis reveals that the component $u$ (with arbitrary mass) of the solution consistently behaves analogous to heat kernel, that is, $u$ exhibiting a Dirac $\delta$ initial singularity identical to that of the fundamental solution, and converges to $0$ in the sense of the $L^p$-norm ($p>1$) as time approaches infinity. However, the initial behavior of the other component $v$ varies significantly based on the value of $\alpha$ and the spatial dimension, exhibiting regularity (not singular), less singularity, or strong singularity (more singular than fundamental solution). Moreover, both $u$ and $v$ undergo instantaneous smoothing, becoming smooth immediately after $t>0$. This phenomenon reveals the adaptive strategies of cells in high-density aggregation environments to prevent resource depletion, reflecting an optimization process of self-organizing behavior.