Boundedness in a quasilinear parabolic–parabolic Keller–Segel system with logistic source

We consider a quasilinear parabolic–parabolic Keller–Segel system involving a source term of logistic type,(0.1) { u t = ∇ ⋅ ( ϕ ( u ) ∇ u ) − ∇ ⋅ ( ψ ( u ) ∇ v ) + g ( u ) , ( x , t ) ∈ Ω × ( 0 , T ) , v t = Δ v − v + u , ( x , t ) ∈ Ω × ( 0 , T ) , with nonnegative initial data under Neumann boundary condition in a smooth bounded domain Ω ⊂ R n , n ⩾ 1 . Here, ϕ and ψ are supposed to be smooth positive functions satisfying c 1 s p ⩽ ϕ and c 1 s q ⩽ ψ ( s ) ⩽ c 2 s q when s ⩾ s 0 with some s 0 > 1 , and we assume that g is smooth on [ 0 , ∞ ) fulfilling g ( 0 ) ⩾ 0 and g ( s ) ⩽ a s − μ s 2 for all s > 0 with constants a ⩾ 0 and μ > 0 . Within this framework, it is proved that whenever q < 1 , for any sufficiently smooth initial data there exists a unique classical solution which is global in time and bounded. Our result is independent of p.