How unstable is spatial homogeneity in Keller-Segel systems? A new critical mass phenomenon in two- and higher-dimensional parabolic-elliptic cases

Abstract

The parabolic-elliptic Keller-Segel system

$$\begin{aligned} \left\{ \begin{array}{ll} u_t = \Delta u - \nabla \cdot (u\nabla v),&{} \\ 0 = \Delta v - \mu + u, &{}\quad \mu :=\frac{1}{|\Omega |} \int _\Omega u, \end{array} \right. \qquad \qquad (\star ) \end{aligned}$$

is considered under homogeneous Neumann boundary conditions in the ball $\Omega =B_R(0)\subset {\mathbb {R}}^n$. The main objective is to reveal that in the context of radially symmetric solutions, this problem exhibits an apparently novel type of critical mass phenomenon: It is shown, namely, that for any choice of $n\ge 2$ and $R>0$ there exists a positive number $m_c=m_c(n,R)$ with the following properties:

• Whenever $m>m_c$, for any nonconstant nonnegative radial initial data $u_0$ with $\int _\Omega u_0=m$ which are, in an appropriately defined sense, more concentrated than the associated spatially homogeneous equilibrium determined by $u\equiv \frac{m}{|\Omega |}$, the corresponding initial-value problem for ($\star$) admits a solution blowing up in finite time; in particular, this implies that any nonconstant and radially nonincreasing initial data $u_0$ with $\int _\Omega u_0>m_c$ enforce blow-up in ($\star$).

• If $m<m_c$, however, then there exist infinitely many nonnegative radial functions $u_0$ which satisfy $\int _\Omega u_0=m$ and which are more concentrated than $u\equiv \frac{m}{|\Omega |}$, but which yet allow for global bounded solutions to ($\star$) emanating from $u_0$.

In consequence, precisely at mass levels above $m_c$ the constant steady states of ($\star$) possess the extreme instability property of repelling arbitrary concentration-increasing perturbations in such a drastic sense that corresponding trajectories collapse in finite time.

Appendix: A comparison principle for (2.6)

Lemma 4.4

Let $L>0$ and $T>0$, and suppose that ${\underline{w}}$ and ${\overline{w}}$ are two functions which belong to $C^1([0,L]\times [0,T))$ and satisfy

$$\begin{aligned} {\underline{w}}_s(s,t)>0 \quad \text{ and } \quad {\overline{w}}(s,t)>0 \quad \text{ for } \text{ all } s\in (0,L) \text{ and } t\in (0,T) \end{aligned}$$

as well as

$$\begin{aligned} {\underline{w}}(\cdot ,t) \in W^{2,\infty }_{loc}((0,L)) \quad \text{ and } \quad {\overline{w}}(\cdot ,t) \in W^{2,\infty }_{loc}((0,L)) \quad \text{ for } \text{ all } t\in (0,T). \end{aligned}$$

If for some constants $a\ge 0, \alpha \in {\mathbb {R}}, b\in {\mathbb {R}}$ and $c\in {\mathbb {R}}$ we have

$$\begin{aligned}&{\underline{w}}_t \le a s^\alpha {\underline{w}}_{ss} + b{\underline{w}}{\underline{w}}_s + c{\underline{w}}_s \quad \text{ and } \quad {\overline{w}}_t \ge a s^\alpha {\overline{w}}_{ss} + b{\overline{w}}{\overline{w}}_s + c{\overline{w}}_s \\&\quad \text{ for } \text{ all } t\in (0,T) \text{ and } \text{ a.e. } s\in (0,L), \end{aligned}$$

and if moreover

$$\begin{aligned} {\underline{w}}(s,0)\le {\overline{w}}(s,0) \quad \text{ for } \text{ all } s\in (0,L) \end{aligned}$$

as well as

$$\begin{aligned} {\underline{w}}(0,t)\le {\overline{w}}(0,t) \quad \text{ and } \quad {\underline{w}}(L,t)\le {\overline{w}}(L,t) \quad \text{ for } \text{ all } t\in (0,T), \end{aligned}$$

then

$$\begin{aligned} {\underline{w}}(s,t)\le {\overline{w}}(s,t) \quad \text{ for } \text{ all } s\in [0,L]\quad \text{ and } \quad t\in [0,T). \end{aligned}$$

Proof

This directly follows upon application of [1, Lemma 5.1]. $\square$