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

Abstract

The quasilinear Keller–Segel system

$$\begin{aligned} \left\{ \begin{array}{l} u_t=\nabla \cdot (D(u)\nabla u) - \nabla \cdot (S(u)\nabla v), \\ v_t=\Delta v-v+u, \end{array}\right. \end{aligned}$$

endowed with homogeneous Neumann boundary conditions is considered in a bounded domain \(\Omega \subset {\mathbb {R}}^n\), \(n \ge 3\), with smooth boundary for sufficiently regular functions D and S satisfying \(D>0\) on \([0,\infty )\), \(S>0\) on \((0,\infty )\) and \(S(0)=0\). On the one hand, it is shown that if \(\frac{S}{D}\) satisfies the subcritical growth condition

$$\begin{aligned} \frac{S(s)}{D(s)} \le C s^\alpha \qquad \text{ for } \text{ all } s\ge 1 \qquad \text{ with } \text{ some } \alpha < \frac{2}{n} \end{aligned}$$

and \(C>0\), then for any sufficiently regular initial data there exists a global weak energy solution such that \({ \mathrm{{ess}}} \sup _{t>0} \Vert u(t) \Vert _{L^p(\Omega )}<\infty \) for some \(p > \frac{2n}{n+2}\). On the other hand, if \(\frac{S}{D}\) satisfies the supercritical growth condition

$$\begin{aligned} \frac{S(s)}{D(s)} \ge c s^\alpha \qquad \text{ for } \text{ all } s\ge 1 \qquad \text{ with } \text{ some } \alpha > \frac{2}{n} \end{aligned}$$

and \(c>0\), 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 \(\alpha = \frac{2}{n}\) for \(n \ge 3\), without any additional assumption on the behavior of D(s) as \(s \rightarrow \infty \), 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 \(Q(s) = \exp (-s^\beta )\), \(s \ge 0\), for global solvability the exponent \(\beta = \frac{n-2}{n}\) is seen to be critical.