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

Abstract

AbstractRadially symmetric global unbounded solutions of the chemotaxis system $$\left\{ {\matrix{{{u_t} = \nabla \cdot (D(u)\nabla u) - \nabla \cdot (uS(u)\nabla v),} \hfill & {} \hfill \cr {0 = \Delta v - \mu + u,} \hfill & {\mu = {1 \over {|\Omega |}}\int_\Omega {u,} } \hfill \cr } } \right.$$ { u t = ∇ ⋅ ( D ( u ) ∇ u ) − ∇ ⋅ ( u S ( u ) ∇ v ) , 0 = Δ v − μ + u , μ = 1 | Ω | ∫ Ω u , are considered in a ball Ω = BR(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 $$m + \lambda < 1 - {2 \over n}$$ m + λ < 1 − 2 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 $${1 \over C} \cdot {(t + 1)^{{1 \over \lambda }}} \le ||u( \cdot ,t)|{|_{{L^\infty }(\Omega )}} \le C \cdot {(t + 1)^{{1 \over \lambda }}}\,\,\,{\rm{for}}\,\,{\rm{all}}\,\,t > 0$$ 1 C ⋅ ( t + 1 ) 1 λ ≤ | | u ( ⋅ , t ) | | L ∞ ( Ω ) ≤ C ⋅ ( t + 1 ) 1 λ f o r a l l t > 0 as well as $$||u( \cdot \,,t)|{|_{{L^1}(\Omega \backslash {B_{{r_0}}}(0))}} \to 0\,\,\,{\rm{as}}\,\,t \to \infty \,\,\,{\rm{for}}\,\,{\rm{all}}\,\,{r_0} \in (0,R)$$ | | u ( ⋅ , t ) | | L 1 ( Ω \ B r 0 ( 0 ) ) → 0 as t → ∞ for all r 0 ∈ ( 0 , R ) with some C > 0.