Exponential grow-up rates in a quasilinear Keller–Segel system

The chemotaxis system ( ⋆ ) u t = ∇ · ( D ( u ) ∇ u ) − ∇ · ( u S ( u ) ∇ v ) , 0 = Δ v − μ + u , μ = 1 | Ω | ∫ Ω u , is considered in a ball Ω = B R ( 0 ) ⊂ R n . It is shown that if S ∈ C 2 ( [ 0 , ∞ ) ) suitably generalizes the prototype given by S ( ξ ) = χ ξ + 1 , ξ ⩾ 0 , with some χ > 0, and if diffusion is suitably weak in the sense that 0 < D ∈ C 2 ( ( 0 , ∞ ) ) is such that there exist K D > 0 and m ∈ ( − ∞ , 1 − 2 n ) fulfilling D ( ξ ) ⩽ K D ξ m − 1 for all ξ > 0 , then for appropriate choices of sufficiently concentrated initial data, an associated no-flux initial-boundary value problem admits a global classical solution ( u , v ) which blows up in infinite time and satisfies 1 C e χ t ⩽ ‖ u ( · , t ) ‖ L ∞ ( Ω ) ⩽ C e χ t for all t > 0 . A major part of the proof is based on a comparison argument involving explicitly constructed subsolutions to a scalar parabolic problem satisfied by mass accumulation functions corresponding to solutions of (⋆).