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 \(m>2(N-1)/N\).


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