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

This paper is concerned with the parabolic-elliptic Keller-Segel system with singular sensitivity and logistic source,{u_t=Δu-χ(u/ vv)+ru-μu²,xεΩ,t>0,0=Δv-v+u,xεΩ, t>0, under homogeneous Neumann boundary conditions in a smoothly bounded domain Ω⊃ R², where χ>0,rεR,μ>0, with nonnegative initial data 0≢ u⁰ε^C0(Ω ̄). 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 rεR. 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 ℓ_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 ℓ_Ωu ^p(x,t)dx can be suitably bounded from above.