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

Abstract We study global solutions of a class of chemotaxis systems generalizing the prototype { u t = ∇ ⋅ ( ( u + 1 ) m − 1 ∇ u ) − χ ∇ ⋅ ( u ( u + 1 ) q − 1 ∇ v ) + a u − b u r , x ∈ Ω , t > 0 , 0 = Δ v − v + u , x ∈ Ω , t > 0 , in a bounded domain Ω ⊂ R N ( N ≥ 1 ) with smooth boundary, with parameters m ≥ 1 , r > 1 , a ≥ 0 , b , q , χ > 0 . It is shown that when q + 1 max ⁡ { r , m + 2 N } , or b > b 0 : = N [ r − m ] − 2 ( r − m ) N + 2 ( r − 2 ) χ if q + 1 = r , then for any sufficiently smooth initial data there exists a classical solution which is global in time and bounded. The results improve the results of Wang et al. (2014) [37] and Cao and Zheng (2014) [6] .