The global-existence-versus-blow-up conjecture for radial fair-competition solutions

Let $\chi>0$, $k\in(-N,0)$, and $m=2N/(2N+k)$ in dimension $N\geq1$. Assume that the initial datum $\rho_0\in{\mathcal Y}$ is radially symmetric. Global-existence-versus-blow-up conjecture. (i) If there exists $\lambda_0>0$ such that

$$\rho_0(r)<\rho_{HLS,\lambda_0,c^*}(r),\qquad\forall r\geq0,$$

then every radially symmetric solution $\rho(t,r)$ of the equation with initial datum $\rho(0,r)=\rho_0(r)$ vanishes in $L^1_{\operatorname{loc}}(\mathbb R^N)$ as $t\to\infty$. (ii) If there exists $\lambda_0>0$ such that

$$\rho_0(r)>\rho_{HLS,\lambda_0,c^*}(r),\qquad\forall r\geq0,$$

then every such solution must blow up at a finite time $T^*$ or develop mass concentration at $r=0$ as $t\to\infty$: there exist $R(t)\to0$ and $C>0$ such that

$$\int_{B(0,R(t))}\rho(t,x)\,dx\geq C.$$

This conjecture describes the expected dichotomy between dispersion below the critical profile and finite-time blow-up or infinite-time concentration above it, extending analogous results in the radial fair-competition setting.