Blow-up prevention by logistic sources in a parabolic–elliptic Keller–Segel system with singular sensitivity

Abstract

This paper is concerned with the parabolic–elliptic Keller–Segel system with singular sensitivity and logistic source,{ut=Δu−χ∇⋅(uv∇v)+ru−μu2,x∈Ω,t>0,0=Δv−v+u,x∈Ω,t>0, under homogeneous Neumann boundary conditions in a smoothly bounded domain Ω⊂R2, where χ>0,r∈R,μ>0, with nonnegative initial data 0≢u0∈C0(Ω̄).It is shown that in this two-dimensional setting, the absorptive character of the logistic kinetics is sufficient to enforce global existence of classical solutions even for arbitrarily large χ>0 and any μ>0 and r∈R. It is moreover shown that if in addition r>0 is sufficiently large then all these solutions are uniformly bounded.A main step in the derivation of these results consists of establishing appropriate positive a priori bounds from below for the mass functional ∫Ωu, which due to the presence of logistic kinetics is not preserved. These in turn provide pointwise lower bounds for v, which then allow for the choice of p>1, explicitly depending inter alia on infv, such that ∫Ωup(x,t)dx can be suitably bounded from above.