A critical exponent in a quasilinear Keller–Segel system with arbitrarily fast decaying diffusivities accounting for volume-filling effects

The quasilinear Keller–Segel system \(\begin{aligned} \left\{ \begin{array}{l} u_t=\nabla \cdot (D(u)\nabla u) - \nabla \cdot (S(u)\nabla v), \\ v_t=\Delta v-v+u, \end{array}\right. \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{c}{u}_t=\mathrm{\nabla }·(D(u)\mathrm{\nabla }u)-\mathrm{\nabla }·(S(u)\mathrm{\nabla }v),\hfill \\ v_t=\mathrm{\Delta }v-v+u,\hfill \end{array}\right)\end{array}$$ endowed with homogeneous Neumann boundary conditions is considered in a bounded domain \(\Omega \subset {\mathbb {R}}^n\) $\mathrm{\Omega }\subset {\mathbb{R}}^n$ , \(n \ge 3\) $n\ge 3$ , with smooth boundary for sufficiently regular functions D and S satisfying \(D>0\) $D>0$ on \([0,\infty )\) $[0,\infty )$ , \(S>0\) $S>0$ on \((0,\infty )\) $(0,\infty )$ and \(S(0)=0\) $S(0)=0$ . On the one hand, it is shown that if \(\frac{S}{D}\) $\frac{S}{D}$ satisfies the subcritical growth condition \(\begin{aligned} \frac{S(s)}{D(s)} \le C s^\alpha \qquad \text{ for } \text{ all } s\ge 1 \qquad \text{ with } \text{ some } \alpha < \frac{2}{n} \end{aligned}\) $$\begin{array}{c}\hfill \frac{S(s)}{D(s)}\le C{s}^{\alpha }\text{for}\text{all}s\ge 1\text{with}\text{some}\alpha <\frac{2}{n}\end{array}$$ and \(C>0\) $C>0$ , then for any sufficiently regular initial data there exists a global weak energy solution such that \({ \mathrm{{ess}}} \sup _{t>0} \Vert u(t) \Vert _{L^p(\Omega )}<\infty \) $\mathrm{ess}{sup}_{t>0}{\Vert u(t)\Vert }_{L^p(\mathrm{\Omega })}<\infty$ for some \(p > \frac{2n}{n+2}\) $p>\frac{2n}{n+2}$ . On the other hand, if \(\frac{S}{D}\) $\frac{S}{D}$ satisfies the supercritical growth condition \(\begin{aligned} \frac{S(s)}{D(s)} \ge c s^\alpha \qquad \text{ for } \text{ all } s\ge 1 \qquad \text{ with } \text{ some } \alpha > \frac{2}{n} \end{aligned}\) $$\begin{array}{c}\hfill \frac{S(s)}{D(s)}\ge c{s}^{\alpha }\text{for}\text{all}s\ge 1\text{with}\text{some}\alpha >\frac{2}{n}\end{array}$$ and \(c>0\) $c>0$ , then the nonexistence of a global weak energy solution having the boundedness property stated above is shown for some initial data in the radial setting. This establishes some criticality of the value \(\alpha = \frac{2}{n}\) $\alpha =\frac{2}{n}$ for \(n \ge 3\) $n\ge 3$ , without any additional assumption on the behavior of D(s) as \(s \rightarrow \infty \) $s\to \infty$ , in particular without requiring any algebraic lower bound for D. When applied to the Keller–Segel system with volume-filling effect for probability distribution functions of the type \(Q(s) = \exp (-s^\beta )\) $Q(s)=exp(-s^{\beta })$ , \(s \ge 0\) $s\ge 0$ , for global solvability the exponent \(\beta = \frac{n-2}{n}\) $\beta =\frac{n-2}{n}$ is seen to be critical.