A family of mass‐critical Keller–Segel systems

Abstract

The no-flux initial-boundary value problem for the quasilinear Keller–Segel system u t = ∇ · ( D ( u ) ∇ u ) − ∇ · ( S ( u ) ∇ v ) , v t = Δ v − 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 ⊂ R n $\Omega \subset \mathbb {R}^n$ , n ⩾ 3 $n\geqslant 3$ , where D ∈ C 2 ( [ 0 , ∞ ) ) $D\in C^2([0,\infty ))$ and S ∈ C 2 ( [ 0 , ∞ ) ) $S\in C^2([0,\infty ))$ are such that D > 0 $D>0$ on [ 0 , ∞ ) $[0,\infty )$ and that S ( 0 ) = 0 < S ( s ) $S(0)=0<S(s)$ for all s > 0 $s>0$ . A particular focus is on cases in which there exist κ > 0 , C S D > 0 $\kappa >0, C_{SD}>0$ and f ∈ L 1 ( ( 1 , ∞ ) ) $f\in L^1((1,\infty ))$ such that − f ( s ) ⩽ D ( s ) S ( s ) − κ s 2 / n ⩽ C S D s for all s ⩾ 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 m 0 > 0 $m_0>0$ such that whenever u 0 $u_0$ and v 0 $v_0$ are reasonably regular and nonnegative with ∫ Ω u 0 < m 0 $\int _\Omega u_0 < m_0$ , within a suitably generalized concept of solvability one can always find a global solution for which u $u$ remains bounded with respect to the norm in L 2 ( n − 1 ) / n ( Ω ) $L^{2(n-1)/{n}}(\Omega )$ . Second, in radially symmetric settings this is complemented by a converse result on nonexistence of such solutions for some appropriately large initial data, hence leading to the conclusion that any pair ( D , S ) $(D,S)$ from the family of nonlinearities fulfilling (**) gives rise to a critical mass phenomenon in (*). Remarkably, this does not only include cases of arbitrarily strong diffusion degeneracies due to fast decay of D ( s ) $D(s)$ as s → ∞ $s\rightarrow \infty$ , but according to the considerably wide funnel described by (*) this moreover indicates that mass criticality in Keller–Segel systems is far more than a nongeneric phenomenon limited to precise functional forms of model ingredients hardly to be found in nature. Applications to concrete scenarios include the detection of mass-critical probability distribution functions of the form 0 ⩽ s ↦ exp ( − s ( n − 2 ) / n ) $0\leqslant s \mapsto \exp (-s^{(n-2)/{n}})$ in the version of (**) accounting for so-called volume-filling effects.