## Application of the best constant of the Sobolev inequality to degenerate Keller-Segel models.(English)Zbl 1171.35068

Global existence and finite time blow-up are investigated for the generalized Smoluchowski-Poisson system (also called the simplified Keller-Segel system with nonlinear diffusion)
$\partial_t u = \nabla\cdot\left( \nabla u^m - \chi u \nabla v \right)\;, \qquad -\Delta v + \gamma v = \alpha u \;\;\;\text{ in }\;\;\; (0,\infty)\times{\mathbb R}^N$
with initial condition $$u(0)=u_0\geq 0$$, the parameters satisfying $$N\geq 3$$, $$m\in (1,2(N-1)/N]$$, $$\chi>0$$, $$\gamma\geq 0$$, and $$\alpha>0$$. It is first shown that there are two explicit positive real numbers $$0<M_1\leq M_2$$ depending on $$N$$, $$\chi$$, and $$\alpha$$ such that: if $$\|u_0\|_1\leq M_1$$, the solution $$(u,v)$$ is global in time, while there are initial data $$u_0$$ such that $$\|u_0\|_1\geq M_2$$ and the corresponding solution $$(u,v)$$ blows up in finite time. Another smallness condition on $$u_0$$ guaranteeing global existence is also provided. It is next proved that, if $$N\geq 3$$ and $$m\in (1,2(N-1)/N)$$, there are initial data $$u_0$$ with an $$L^{N(2-m)/2}$$-norm exceeding an explicit threshold value for which the corresponding solution $$(u,v)$$ blows up in finite time. Recall that it has already been established that no finite time blow-up occurs when $$N\geq 3$$ and $$m>2(N-1)/N$$.

### MSC:

 35K65 Degenerate parabolic equations 35K45 Initial value problems for second-order parabolic systems 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 35B35 Stability in context of PDEs