Complete infinite-time mass aggregation in a quasilinear Keller–Segel system

Abstract

Radially symmetric global unbounded solutions of the chemotaxis system

$$\left\{ {\matrix{{{u_t} = \nabla \cdot (D(u)\nabla u) - \nabla \cdot (uS(u)\nabla v),} \hfill & {} \hfill \cr {0 = \Delta v - \mu + u,} \hfill & {\mu = {1 \over {|\Omega |}}\int_\Omega {u,} } \hfill \cr } } \right.$$

are considered in a ball Ω = B_R(0) ⊂ ℝⁿ, where n ≥ 3 and R > 0.

Under the assumption that D and S suitably generalize the prototypes given by D(ξ) = (ξ + ι)^m−1 and S(ξ) = (ξ + 1)^−λ−1 for all ξ > 0 and some m ∈ ℝ, λ >0 and ι ≥ 0 fulfilling \(m + \lambda < 1 - {2 \over n}\), a considerably large set of initial data u₀ is found to enforce a complete mass aggregation in infinite time in the sense that for any such u₀, an associated Neumann type initial-boundary value problem admits a global classical solution (u, v) satisfying

$${1 \over C} \cdot {(t + 1)^{{1 \over \lambda }}} \le ||u( \cdot ,t)|{|_{{L^\infty }(\Omega )}} \le C \cdot {(t + 1)^{{1 \over \lambda }}}\,\,\,{\rm{for}}\,\,{\rm{all}}\,\,t > 0$$

as well as

$$||u( \cdot \,,t)|{|_{{L^1}(\Omega \backslash {B_{{r_0}}}(0))}} \to 0\,\,\,{\rm{as}}\,\,t \to \infty \,\,\,{\rm{for}}\,\,{\rm{all}}\,\,{r_0} \in (0,R)$$

with some C > 0.