Finite time blowup for Keller-Segel equation with logistic damping in three dimensions
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arXiv
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| Autores principales: | , , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866916873145155584 |
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| author | Liu, Jiaqi Wang, Yixuan Zhou, Tao |
| author_facet | Liu, Jiaqi Wang, Yixuan Zhou, Tao |
| contents | The Keller-Segel equation, a classical chemotaxis model, and many of its variants have been extensively studied for decades. In this work, we focus on 3D Keller-Segel equation with a quadratic logistic damping term $-μρ^2$ (modeling density-dependent mortality rate) and show the existence of finite-time blowup solutions with nonnegative density and finite mass for any $μ\in \big[0,\frac{1}{3}\big)$. This range of $μ$ is sharp; for $μ\ge \frac{1}{3}$, the logistic damping effect suppresses the blowup as shown in [Kang-Stevens, 2016] and [Tello-Winkler, 2007]. A key ingredient is to construct a self-similar blowup solution to a related aggregation equation as an approximate solution, with subcritical scaling relative to the original model. Based on this construction, we employ a robust weighted $L^2$ method to prove the stability of this approximate solution, where modulation ODEs are introduced to enforce local vanishing conditions for the perturbation lying in a singular-weighted $L^2$ space. As a byproduct, we exhibit a new family of type I blowup mechanisms for the classical 3D Keller-Segel equation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_12231 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Finite time blowup for Keller-Segel equation with logistic damping in three dimensions Liu, Jiaqi Wang, Yixuan Zhou, Tao Analysis of PDEs The Keller-Segel equation, a classical chemotaxis model, and many of its variants have been extensively studied for decades. In this work, we focus on 3D Keller-Segel equation with a quadratic logistic damping term $-μρ^2$ (modeling density-dependent mortality rate) and show the existence of finite-time blowup solutions with nonnegative density and finite mass for any $μ\in \big[0,\frac{1}{3}\big)$. This range of $μ$ is sharp; for $μ\ge \frac{1}{3}$, the logistic damping effect suppresses the blowup as shown in [Kang-Stevens, 2016] and [Tello-Winkler, 2007]. A key ingredient is to construct a self-similar blowup solution to a related aggregation equation as an approximate solution, with subcritical scaling relative to the original model. Based on this construction, we employ a robust weighted $L^2$ method to prove the stability of this approximate solution, where modulation ODEs are introduced to enforce local vanishing conditions for the perturbation lying in a singular-weighted $L^2$ space. As a byproduct, we exhibit a new family of type I blowup mechanisms for the classical 3D Keller-Segel equation. |
| title | Finite time blowup for Keller-Segel equation with logistic damping in three dimensions |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2504.12231 |