Infinitely many self-similar blow-up profiles for the Keller-Segel system in dimensions 3 to 9
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866929752487493632 |
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| author | Nguyen, Van Tien Wang, Zhi-An Zhang, Kaiqiang |
| author_facet | Nguyen, Van Tien Wang, Zhi-An Zhang, Kaiqiang |
| contents | Based on the method of matched asymptotic expansions and Banach fixed point theorem, we rigorously construct infinitely many self-similar blow-up profiles for the parabolic-elliptic Keller-Segel system \begin{equation*} \left\{\begin{array}{l} \partial_{t} u=Δu-\nabla \cdot\left(u \nabla Φ_{u}\right), \\ 0=ΔΦ_{u}+u,\\ u(\cdot,0)=u_0 \geq 0 \end{array}\quad \text{in}\ \mathbb{R}^{d},\right. \end{equation*} where $d\in \{3,\cdots,9\}$. Our findings demonstrate that the infinitely many backward self-similar profiles approximate the rescaling radial steady-state near the origin (i.e. $0<|x|\ll1$) and $\frac{2(d-2)}{|x|^2}$ at spatial infinity (i.e. $|x|\gg1$). We also establish the convergence of the self-similar blow-up solutions as time tends to the blow-up time $T>0$. Our results can give a refined description of backward self-similar profiles for all $|x|\geq 0$ rather than for $0<|x|\ll1$ or $|x|\gg1$, indicating that the blow-up point is the origin and $$ u(x,t)\sim \frac{1}{|x|^2},\ \ \ x\ne0,\ \text{as}\ t\to T. $$ |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_02263 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Infinitely many self-similar blow-up profiles for the Keller-Segel system in dimensions 3 to 9 Nguyen, Van Tien Wang, Zhi-An Zhang, Kaiqiang Analysis of PDEs 35B44, 35C06, 35K57, 35Q92 Based on the method of matched asymptotic expansions and Banach fixed point theorem, we rigorously construct infinitely many self-similar blow-up profiles for the parabolic-elliptic Keller-Segel system \begin{equation*} \left\{\begin{array}{l} \partial_{t} u=Δu-\nabla \cdot\left(u \nabla Φ_{u}\right), \\ 0=ΔΦ_{u}+u,\\ u(\cdot,0)=u_0 \geq 0 \end{array}\quad \text{in}\ \mathbb{R}^{d},\right. \end{equation*} where $d\in \{3,\cdots,9\}$. Our findings demonstrate that the infinitely many backward self-similar profiles approximate the rescaling radial steady-state near the origin (i.e. $0<|x|\ll1$) and $\frac{2(d-2)}{|x|^2}$ at spatial infinity (i.e. $|x|\gg1$). We also establish the convergence of the self-similar blow-up solutions as time tends to the blow-up time $T>0$. Our results can give a refined description of backward self-similar profiles for all $|x|\geq 0$ rather than for $0<|x|\ll1$ or $|x|\gg1$, indicating that the blow-up point is the origin and $$ u(x,t)\sim \frac{1}{|x|^2},\ \ \ x\ne0,\ \text{as}\ t\to T. $$ |
| title | Infinitely many self-similar blow-up profiles for the Keller-Segel system in dimensions 3 to 9 |
| topic | Analysis of PDEs 35B44, 35C06, 35K57, 35Q92 |
| url | https://arxiv.org/abs/2503.02263 |