Boundedness of solutions to a quasilinear parabolic–elliptic Keller–Segel system with logistic source

We study a quasilinear parabolic–elliptic Keller–Segel system involving a source term of logistic type u t  = ∇ ⋅ ( ϕ ( u ) ∇  u ) −  χ  ∇ ⋅ ( u  ∇  v ) +  g ( u ), − Δ v  = −  v  +  u in Ω × (0, T ), subject to nonnegative initial data and the homogeneous Neumann boundary condition in a bounded domain Ω ⊂ R nwith smooth boundary, n  ≥ 1, χ  > 0, ϕ  ≥  c 1 s p for s  ≥  s 0  > 1, and g ( s ) ≤  as  −  μs 2 for s  > 0 with a , g (0) ≥ 0, μ  > 0. There are three nonlinear mechanisms included in the chemotaxis model: the nonlinear diffusion, aggregation and logistic absorption. The interaction among the triple nonlinearities shows that together with the nonlinear diffusion, the logistic absorption will dominate the aggregation such that the unique classical solution of the system has to be global in time and bounded, regardless of the initial data, whenever μ > χ 1 −2 n ( 1 − p )+, or, equivalently, p > 1 − 2 χ n( χ − μ )+, which enlarge the parameter range μ > n − 2 n χ , or p > 1 − 2 n , required by globally bounded solutions of the quasilinear K‐S system without the logistic source. Copyright © 2013 John Wiley & Sons, Ltd.