Finite Time Blow-Up and Chemotactic Collapse in Keller–Segel Model with Signal Consumption

IF 2.6 2区数学 Q1 MATHEMATICS, APPLIED

Journal of Nonlinear Science Pub Date : 2024-06-10 DOI:10.1007/s00332-024-10045-3

Chunhua Jin

引用次数: 0

Abstract

The occurrence of finite time blow-up phenomenon in the Keller–Segel (KS) model has always been a significant area of interest for mathematicians. Despite extensive research on the blow-up phenomenon in KS models with signal production, Understanding of this phenomenon in models with signal consumption mechanisms has been scarce.This paper marks a preliminary investigation into this unexplored field. In this study, we employ a backward self-similar solution to demonstrate that the finite time blowup indeed occurs within this model. More precisely, in one-dimensional space, finite time blowing up corresponding to the chemotactic collapse phenomenon (the formation of Dirac \(\delta \)-singularity ) happens; in high-dimensional space, the self-similar solution will blow up everywhere. Finally, we also consider the special cases where the diffusion coefficient of bacteria or oxygen is 0. For these cases, chemotactic collapse phenomenon occurs in both one-dimensional and two-dimensional spaces.

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带有信号消耗的凯勒-西格尔模型中的有限时间膨胀和趋化崩溃

凯勒-西格尔（Keller-Segel，KS）模型中出现的有限时间炸毁现象一直是数学家们感兴趣的一个重要领域。尽管对具有信号产生机制的 KS 模型中的炸裂现象进行了大量研究，但对具有信号消耗机制的模型中的炸裂现象的了解却很少。在这项研究中，我们采用了一种后向自相似解来证明有限时间炸毁现象确实发生在该模型中。更准确地说，在一维空间中，有限时间炸裂对应于趋化坍缩现象（形成狄拉克（\delta \）奇异性）；在高维空间中，自相似解将处处炸裂。最后，我们还考虑了细菌或氧气的扩散系数为 0 的特殊情况，对于这些情况，趋化坍缩现象在一维和二维空间都会发生。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Nonlinear Science 数学-力学

CiteScore

5.00

自引率

3.30%

发文量

审稿时长

4.5 months

期刊介绍： The mission of the Journal of Nonlinear Science is to publish papers that augment the fundamental ways we describe, model, and predict nonlinear phenomena. Papers should make an original contribution to at least one technical area and should in addition illuminate issues beyond that area''s boundaries. Even excellent papers in a narrow field of interest are not appropriate for the journal. Papers can be oriented toward theory, experimentation, algorithms, numerical simulations, or applications as long as the work is creative and sound. Excessively theoretical work in which the application to natural phenomena is not apparent (at least through similar techniques) or in which the development of fundamental methodologies is not present is probably not appropriate. In turn, papers oriented toward experimentation, numerical simulations, or applications must not simply report results without an indication of what a theoretical explanation might be. All papers should be submitted in English and must meet common standards of usage and grammar. In addition, because ours is a multidisciplinary subject, at minimum the introduction to the paper should be readable to a broad range of scientists and not only to specialists in the subject area. The scientific importance of the paper and its conclusions should be made clear in the introduction-this means that not only should the problem you study be presented, but its historical background, its relevance to science and technology, the specific phenomena it can be used to describe or investigate, and the outstanding open issues related to it should be explained. Failure to achieve this could disqualify the paper.