A family of mass‐critical Keller–Segel systems

The no‐flux initial‐boundary value problem for the quasilinear Keller–Segel system * ut=∇·(D(u)∇u)−∇·(S(u)∇v),vt=Δv−v+u,\begin{equation} \hspace*{6pc}{\left\lbrace \def\eqcellsep{&}\begin{array}{l}u_t=\nabla \cdot (D(u)\nabla u) - \nabla \cdot (S(u)\nabla v), \hspace*{-6pc}\\[3pt] v_t=\Delta v-v+u, \end{array} \right.} \end{equation}is considered in smoothly bounded domains Ω⊂Rn$\Omega \subset \mathbb {R}^n$ , n⩾3$n\geqslant 3$ , where D∈C2([0,∞))$D\in C^2([0,\infty ))$ and S∈C2([0,∞))$S\in C^2([0,\infty ))$ are such that D>0$D>0$ on [0,∞)$[0,\infty )$ and that S(0)=00$s>0$ . A particular focus is on cases in which there exist κ>0,CSD>0$\kappa >0, C_{SD}>0$ and f∈L1((1,∞))$f\in L^1((1,\infty ))$ such that ** −f(s)⩽D(s)S(s)−κs2/n⩽CSDsforalls⩾1.\begin{equation} \hspace*{6pc}- f(s) \leqslant \frac{D(s)}{S(s)} - \frac{\kappa }{s^{2/n}} \leqslant \frac{C_{SD}}{s} \quad \mbox{for all } s\geqslant 1.\hspace*{-6pc} \end{equation}It is first shown that then there exists m0>0$m_0>0$ such that whenever u0$u_0$ and v0$v_0$ are reasonably regular and nonnegative with ∫Ωu0