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

Abstract

AbstractThe quasilinear Keller–Segel system $$\begin{aligned} \left\{ \begin{array}{l} u_t=\nabla \cdot (D(u)\nabla u) - \nabla \cdot (S(u)\nabla v), \\ v_t=\Delta v-v+u, \end{array}\right. \end{aligned}$$ u t = ∇ · ( D ( u ) ∇ u ) - ∇ · ( S ( u ) ∇ v ) , v t = Δ v - v + u , endowed with homogeneous Neumann boundary conditions is considered in a bounded domain $$\Omega \subset {\mathbb {R}}^n$$ Ω ⊂ R n , $$n \ge 3$$ n ≥ 3 , with smooth boundary for sufficiently regular functions D and S satisfying $$D>0$$ D > 0 on $$[0,\infty )$$ [ 0 , ∞ ) , $$S>0$$ S > 0 on $$(0,\infty )$$ ( 0 , ∞ ) and $$S(0)=0$$ S ( 0 ) = 0 . On the one hand, it is shown that if $$\frac{S}{D}$$ S D satisfies the subcritical growth condition $$\begin{aligned} \frac{S(s)}{D(s)} \le C s^\alpha \qquad \text{ for } \text{ all } s\ge 1 \qquad \text{ with } \text{ some } \alpha < \frac{2}{n} \end{aligned}$$ S ( s ) D ( s ) ≤ C s α for all s ≥ 1 with some α < 2 n and $$C>0$$ C > 0 , then for any sufficiently regular initial data there exists a global weak energy solution such that $${ \mathrm{{ess}}} \sup _{t>0} \Vert u(t) \Vert _{L^p(\Omega )}<\infty $$ ess sup t > 0 ‖ u ( t ) ‖ L p ( Ω ) < ∞ for some $$p > \frac{2n}{n+2}$$ p > 2 n n + 2 . On the other hand, if $$\frac{S}{D}$$ S D satisfies the supercritical growth condition $$\begin{aligned} \frac{S(s)}{D(s)} \ge c s^\alpha \qquad \text{ for } \text{ all } s\ge 1 \qquad \text{ with } \text{ some } \alpha > \frac{2}{n} \end{aligned}$$ S ( s ) D ( s ) ≥ c s α for all s ≥ 1 with some α > 2 n 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 $$\alpha = \frac{2}{n}$$ α = 2 n for $$n \ge 3$$ n ≥ 3 , without any additional assumption on the behavior of D(s) as $$s \rightarrow \infty $$ s → ∞ , 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 β ) , $$s \ge 0$$ s ≥ 0 , for global solvability the exponent $$\beta = \frac{n-2}{n}$$ β = n - 2 n is seen to be critical.