Global bounded solutions in a two-dimensional quasilinear Keller–Segel system with exponentially decaying diffusivity and subcritical sensitivity

Abstract

The quasilinear chemotaxis system (∗){ut=∇⋅(D(u)∇u)−∇⋅(S(u)∇v),vt=Δv−v+u, is considered under homogeneous Neumann boundary conditions in a bounded domain Ω⊂R2 with smooth boundary.It is shown that if D and S are sufficiently smooth nonnegative functions on [0,∞) satisfying K1e−β−s≤D(s)≤K2e−β+sfor all s≥0 with some K1>0,K2>0, β+>0 and β−≥β+, then whenever S satisfies the condition of subcritical growth relative to D given by S(s)D(s)≤K3sαfor all s≥0 with some K3>0 and α∈(0,1), for all suitably regular nonnegative initial data the corresponding initial–boundary value problem for (⋆) possesses a global classical solution for which the component u is bounded in Ω×(0,∞).