Shrinking vs. expanding: the evolution of spatial support in degenerate Keller-Segel systems

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=\Delta v -\mu +u,\quad\mu =\frac{1}{|\Omega|}\int_\Omega u, \end{cases} \end{align*} in balls $\Omega\subset\mathbb R^n$, $n\ge 1$, 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, H\"older continuous) initial data $u_0$ near the boundary of its support $\overline{B_{r_1}(0)}\subsetneq\Omega$: 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 $\int_\Omega u_0$) 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 $\zeta>0$ such that $\sup\{\, |x| \mid x \in \operatorname{supp} u(\cdot, t)\,\}\le r_1 -\zeta t$ 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 $\zeta>0$ such that $\sup\{\, |x| \mid x \in \operatorname{supp} u(\cdot, t)\,\}\ge r_1 +\zeta t$ for all $t\in(0, T)$.