| Abstract: |
| We study radially symmetric solutions to the parabolic-elliptic Keller-Segel system of the form
\begin{align*}
\left\lbrace
\begin{array}{r@{}l@{\quad}l}
&u_t=\Delta u-\nabla \cdot \big(u\nabla v\big),\
&0=\Delta v -\mu + u, \qquad \mu=\frac1{|\Omega|}\int_\Omega u_0,\
\end{array}\right.(\star)
\end{align*}
with $u_0 \in C^0(\overline\Omega)$, subjected to homogeneous Neumann boundary conditions and posed in the domain $\Omega=B_R(0)\subset \mathbb R^n$ for some $R>0$ and $n\geq 3$.
In two dimensions, it is well-known that solutions blowing up in finite time converge to a Dirac profile in the vague topology. In dimensions greater or equal to three, there are cases in which Dirac singularities do form, although at least for radially decreasing initial data only some time after blow-up.
Examining the system
\begin{align*}
\left\lbrace
\begin{array}{r@{}l@{\quad}l}
&w_t=n^2 s^{2-\frac2n} w_{ss} + n ww_s - \mu s w_s, \qquad& s\in(0,R^n),~t>0,\
&w(R^n,t)=\frac{\mu R^n}{n}, \qquad& t>0,
\end{array}\right.
\end{align*}
emerging from a transformation of $(\star)$, specifically $w(s,t)=\int_0^{s^\frac1n} \rho^{n-1} u(\rho,t) d\rho$, we may trace the evolution of solutions further than the blow-up time of $(\star)$ and thus analyze the behavior of the Dirac singularity corresponding to $w(0,t)>0$. In particular, we shall focus on the monotonicity and long-term behavior of $t \mapsto w(0,t)$. |
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