Shrinking vs. expanding: the evolution of spatial support in degenerate Keller-Segel systems
Mario Fuest, Frederic Heihoff
TL;DR
The paper analyzes the initial evolution of the positivity set for radially symmetric solutions of a degenerate Keller–Segel system in a ball, identifying a sharp boundary-flattening threshold $A_{\mathrm{crit}}$ that separates inward from outward motion of the outer free boundary. By transforming to a mass-accumulation function $w(s,t)$ and employing a comparison principle with carefully constructed sub- and supersolutions (regularized as needed), the authors reduce the problem to the scalar dynamics of $w$ and derive explicit near-boundary criteria involving $u_0(x) \sim (r_1-|x|)^{1/(m-1)}$. The main result establishes that initial data flatter than the threshold shrink the support, while steeper data expand it, at least for short times. This work advances the understanding of finite-speed propagation and boundary behavior in degenerate chemotaxis systems, highlighting the delicate balance between degenerate diffusion and chemotactic aggregation near the boundary.
Abstract
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 $t\in(0, T)$.