## Self-similar blow up with a continuous range of values of the aggregated mass for a degenerate Keller-Segel system.(English)Zbl 1221.35087

The existence of self-similar blowing-up (or backward) solutions with finite mass is investigated for the degenerate parabolic-elliptic Keller-Segel system
$\partial_t u = \nabla\cdot (\nabla u^m - u \nabla v), \quad 0 = \Delta v + u, \quad (x,t)\in {\mathbb R}^N\times (0,\infty),$
with critical diffusion exponent $$m=2(N-1)/N$$ and $$N\geq 3$$. More precisely, given $$T>0$$, solutions of the form
$u(x,t) = \frac{1}{T-t} \Phi\left( \frac{|x|}{(T-t)^{1/N}} \right), \quad v(x,t) = \frac{1}{(T-t)^{(N-2)/N}} \Psi\left( \frac{|x|}{(T-t)^{1/N}} \right),$
for $$(x,t)\in {\mathbb R}^N\times (0,T)$$ are sought for some nonnegative and integrable function $$\Phi$$ and nonnegative function $$\Psi$$. The profiles $$\Phi$$ and $$\Psi$$ are then given by
$\Phi_\beta(r) = \left( \frac{N-2}{2(N-1)} \right)^{N/(N-2)} Q_\beta(r)^{N/(N-2)}, \quad \Psi_\beta(r) = Q_\beta(r) - \frac{r^2}{2N},$
for some $$\beta>0$$, and $$Q_\beta$$ solves the initial-value problem
$Q_\beta''(r) + \frac{N-1}{r} Q_\beta'(r) + \left( \frac{N-2}{2(N-1)} \right)^{N/(N-2)} Q_\beta(r)^{N/(N-2)} - 1 = 0, \quad Q_\beta(0)=0,$
as long as $$Q_\beta(r)>0$$. It is proved that, for $$\beta$$ large enough, $$Q_\beta$$ indeed vanishes at some point $$r_\beta>0$$ and gives rise to a compactly supported profile $$\Phi_\beta$$ with finite mass $$m_\beta=\|\Phi_\beta\|_1$$. First-order expansions of $$r_\beta$$ and $$m_\beta$$ as $$\beta\to\infty$$ are computed, showing that $$m_\beta$$ converges to the previously identified threshold mass $$m_\infty$$ below which no finite time blow-up phenomenon can occur. A complete study of the behaviour of $$Q_\beta$$ for all values of $$\beta>0$$ is performed in [A. Blanchet and Ph. Laurençot, “Finite mass self-similar blowing-up solution of a chemotaxis system with non-linear diffusion”, Commun. Pure Appl. Anal., to appear].

### MSC:

 35C06 Self-similar solutions to PDEs 35K45 Initial value problems for second-order parabolic systems 35B44 Blow-up in context of PDEs 35K65 Degenerate parabolic equations 92C17 Cell movement (chemotaxis, etc.)