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, Hölder 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 $u_0(x)\le A(r_1-|x|)^\frac{1}{m-1}$ for all $|x|\in(r_0, r_1)$ and some $r_0\in(0,r_1)$ and $A \lt A_{\mathrm{crit}}$ then there are T > 0 and ζ > 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 $u_0(x)\ge A(r_1-|x|)^\frac{1}{m-1}$ for all $|x|\in(r_0, r_1)$ and some $r_0 \in (0, r_1)$ and $A \gt A_{\mathrm{crit}}$ then we can find T > 0 and ζ > 0 such that $\sup\{\, |x| \mid x \in \operatorname{supp} u(\cdot, t)\,\}\ge r_1 +\zeta t$ for all $t\in(0, T)$ .