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=Δv -μ+u,\quadμ=\frac{1}{|Ω|}\int_Ωu, \end{cases} \end{align*} in balls $\hat{I}\text{c◯}\subset {\mathbb{R}}^n$, $n\ge 1$, 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 $\overline{B_{r_1}(0)}⊊\hat{I}\text{c◯}$: 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 ${\int }_{\hat{I}}\text{c◯}{u}_0$) 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 $\hat{I}\P >0$ such that $\sup \{|x|\mid x\in \mathrm{supp}u(\cdot ,t)\}\le r_1-\hat{I}\P t$ 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 $\hat{I}\P >0$ such that $\sup \{|x|\mid x\in \mathrm{supp}u(\cdot ,t)\}\ge r_1+\hat{I}\P t$ for all $t\in (0,T)$.

18 pages