Shrinking vs. expanding: the evolution of spatial support in degenerate Keller-Segel systems
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| Format: | Preprint |
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2025
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| _version_ | 1866911366790512640 |
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| author | Fuest, Mario Heihoff, Frederic |
| author_facet | Fuest, Mario Heihoff, Frederic |
| contents | We consider radially symmetric solutions of the degenerate Keller-Segel system \begin{align*}
\begin{cases}
\partial_t u=\nabla\cdot (u^{m-1}\nabla u - u\nabla v),\\
0=Δv -μ+u,\quadμ=\frac{1}{|Ω|}\int_Ωu,
\end{cases} \end{align*} in balls $Ω\subset\mathbb R^n$, $n\ge 1$, where $m>1$ is arbitrary. Our main result states that the initial evolution of the positivity set of $u$ is essentially determined by the shape of the (nonnegative, radially symmetric, Hölder continuous) initial data $u_0$ near the boundary of its support $\overline{B_{r_1}(0)}\subsetneqΩ$: It shrinks for sufficiently flat and expands for sufficiently steep $u_0$.
More precisely, there exists an explicit constant $A_{\mathrm{crit}} \in (0, \infty)$ (depending only on $m, n, R, r_1$ and $\int_Ωu_0$) such that if \begin{align*}
u_0(x)\le A(r_1-|x|)^\frac{1}{m-1}
\qquad \text{for all $|x|\in(r_0, r_1)$ and some $r_0\in(0,r_1)$ and $A<A_{\mathrm{crit}}$}, \end{align*} then there are $T>0$ and $ζ>0$ such that $\sup\{\, |x| \mid x \in \operatorname{supp} u(\cdot, t)\,\}\le r_1 -ζt$ for all $t\in(0, T)$, while if \begin{align*}
u_0(x)\ge A(r_1-|x|)^\frac{1}{m-1}
\qquad \text{for all $|x|\in(r_0, r_1)$ and some $r_0 \in (0, r_1)$ and $A>A_{\mathrm{crit}}$}, \end{align*} then we can find $T>0$ and $ζ>0$ such that $\sup\{\, |x| \mid x \in \operatorname{supp} u(\cdot, t)\,\}\ge r_1 +ζt$ for all $t\in(0, T)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_19119 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Shrinking vs. expanding: the evolution of spatial support in degenerate Keller-Segel systems Fuest, Mario Heihoff, Frederic Analysis of PDEs 35B33 (primary), 35B51, 35K59, 35K65, 92C17 (secondary) We consider radially symmetric solutions of the degenerate Keller-Segel system \begin{align*} \begin{cases} \partial_t u=\nabla\cdot (u^{m-1}\nabla u - u\nabla v),\\ 0=Δv -μ+u,\quadμ=\frac{1}{|Ω|}\int_Ωu, \end{cases} \end{align*} in balls $Ω\subset\mathbb R^n$, $n\ge 1$, where $m>1$ is arbitrary. Our main result states that the initial evolution of the positivity set of $u$ is essentially determined by the shape of the (nonnegative, radially symmetric, Hölder continuous) initial data $u_0$ near the boundary of its support $\overline{B_{r_1}(0)}\subsetneqΩ$: It shrinks for sufficiently flat and expands for sufficiently steep $u_0$. More precisely, there exists an explicit constant $A_{\mathrm{crit}} \in (0, \infty)$ (depending only on $m, n, R, r_1$ and $\int_Ωu_0$) such that if \begin{align*} u_0(x)\le A(r_1-|x|)^\frac{1}{m-1} \qquad \text{for all $|x|\in(r_0, r_1)$ and some $r_0\in(0,r_1)$ and $A<A_{\mathrm{crit}}$}, \end{align*} then there are $T>0$ and $ζ>0$ such that $\sup\{\, |x| \mid x \in \operatorname{supp} u(\cdot, t)\,\}\le r_1 -ζt$ for all $t\in(0, T)$, while if \begin{align*} u_0(x)\ge A(r_1-|x|)^\frac{1}{m-1} \qquad \text{for all $|x|\in(r_0, r_1)$ and some $r_0 \in (0, r_1)$ and $A>A_{\mathrm{crit}}$}, \end{align*} then we can find $T>0$ and $ζ>0$ such that $\sup\{\, |x| \mid x \in \operatorname{supp} u(\cdot, t)\,\}\ge r_1 +ζt$ for all $t\in(0, T)$. |
| title | Shrinking vs. expanding: the evolution of spatial support in degenerate Keller-Segel systems |
| topic | Analysis of PDEs 35B33 (primary), 35B51, 35K59, 35K65, 92C17 (secondary) |
| url | https://arxiv.org/abs/2501.19119 |