Slow Grow-up in a Quasilinear Keller–Segel System

Abstract

AbstractIn a ball $$\Omega =B_R(0)\subset \mathbb {R}^n$$ Ω = B R ( 0 ) ⊂ R n , $$n\ge 2$$ n ≥ 2 , the chemotaxis system $$\begin{aligned} \left\{ \begin{array}{l}u_t = \nabla \cdot \big ( D(u) \nabla u \big ) - \nabla \cdot \big ( uS(u)\nabla v\big ), \\ 0 = \Delta v - \mu + u, \qquad \mu =\frac{1}{|\Omega |} \int _\Omega u, \end{array} \right. \qquad \qquad (\star ) \end{aligned}$$ u t = ∇ · ( D ( u ) ∇ u ) - ∇ · ( u S ( u ) ∇ v ) , 0 = Δ v - μ + u , μ = 1 | Ω | ∫ Ω u , ( ⋆ ) is considered under no-flux boundary conditions, with a focus on nonlinearities $$S\in C^2([0,\infty ))$$ S ∈ C 2 ( [ 0 , ∞ ) ) which exhibit super-algebraically fast decay in the sense that with some $$K_S>0, \beta \in [0,1)$$ K S > 0 , β ∈ [ 0 , 1 ) and $$\xi _0>0$$ ξ 0 > 0 , $$\begin{aligned} S(\xi )>0 \quad \text{ and } \quad S'(\xi ) \le -K_S\xi ^{-\beta } S(\xi ) \qquad \text{ for } \text{ all } \xi \ge \xi _0. \end{aligned}$$ S ( ξ ) > 0 and S ′ ( ξ ) ≤ - K S ξ - β S ( ξ ) for all ξ ≥ ξ 0 . It is, inter alia, shown that if furthermore $$D\in C^2((0,\infty ))$$ D ∈ C 2 ( ( 0 , ∞ ) ) is positive and suitably small in relation to S by satisfying $$\begin{aligned} \frac{\xi S(\xi )}{D(\xi )} \ge K_{SD}\xi ^\lambda \qquad \text{ for } \text{ all } \xi \ge \xi _0 \end{aligned}$$ ξ S ( ξ ) D ( ξ ) ≥ K SD ξ λ for all ξ ≥ ξ 0 with some $$K_{SD}>0$$ K SD > 0 and $$\lambda >\frac{2}{n}$$ λ > 2 n , then throughout a considerably large set of initial data, ($$\star $$ ⋆ ) admits global classical solutions (u, v) fulfilling $$\begin{aligned} \frac{z(t)}{C} \le \Vert u(\cdot ,t)\Vert _{L^\infty (\Omega )} \le Cz(t) \qquad \text{ for } \text{ all } t>0, \end{aligned}$$ z ( t ) C ≤ ‖ u ( · , t ) ‖ L ∞ ( Ω ) ≤ C z ( t ) for all t > 0 , with some $$C=C^{(u,v)}\ge 1$$ C = C ( u , v ) ≥ 1 , where z denotes the solution of $$\begin{aligned} \left\{ \begin{array}{l}z'(t) = z^2(t) \cdot S\big ( z(t)\big ), \qquad t>0, \\ z(0)=\xi _0, \end{array} \right. \end{aligned}$$ z ′ ( t ) = z 2 ( t ) · S ( z ( t ) ) , t > 0 , z ( 0 ) = ξ 0 , which is seen to exist globally, and to satisfy $$z(t)\rightarrow +\infty $$ z ( t ) → + ∞ as $$t\rightarrow \infty $$ t → ∞ . As particular examples, exponentially and doubly exponentially decaying S are found to imply corresponding infinite-time blow-up properties in ($$\star $$ ⋆ ) at logarithmic and doubly logarithmic rates, respectively.