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| Format: | Preprint |
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2026
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| Accès en ligne: | https://arxiv.org/abs/2603.19663 |
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| _version_ | 1866912975483305984 |
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| author | Jin, Chunhua |
| author_facet | Jin, Chunhua |
| contents | In this paper, we concentrate on investigating the self-similar singular solutions of Keller-Segel model with signal consumption ($-uv^α$) 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 $α$ 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 $δ$ 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 $α$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_19663 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Classification of self-similar singular solutions with large mass for Keller-Segel model with signal consumption Jin, Chunhua Analysis of PDEs 35C06, 35K67, 92C17 In this paper, we concentrate on investigating the self-similar singular solutions of Keller-Segel model with signal consumption ($-uv^α$) 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 $α$ 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 $δ$ 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 $α$ 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. |
| title | Classification of self-similar singular solutions with large mass for Keller-Segel model with signal consumption |
| topic | Analysis of PDEs 35C06, 35K67, 92C17 |
| url | https://arxiv.org/abs/2603.19663 |