Boundedness in a quasilinear chemotaxis‐haptotaxis system with logistic source

This paper studies the chemotaxis‐haptotaxis system with nonlinear diffusionu t = ∇ · ( D ( u ) ∇ u ) − χ ∇ · ( u ∇ v ) − ξ ∇ · ( u ∇ w ) + μ u ( 1 − u − w ) ,x ∈ Ω ,t > 0 ,0 = Δ v − v + u ,x ∈ Ω ,t > 0 ,w t = − v w ,x ∈ Ω ,t > 0 ,subject to the homogeneous Neumann boundary conditions and suitable initial conditions, where χ , ξ and μ are positive constants, and Ω ⊂ R n ( n ⩾2) is a bounded and smooth domain. Here, we assume that D ( u )⩾ c D u m  − 1 for all u  > 0 with some c D  > 0 and m ⩾1. For the case of non‐degenerate diffusion, if μ  >  μ ∗ , whereμ ∗ : =( 2 − m ) n − 2 ( 2 − m ) n χ ,if m < 2 − 2 n ,0 ,if m ⩾ 2 − 2 n ,it is proved that the system possesses global classical solutions which are uniformly‐in‐time bounded. In the case of degenerate diffusion, we show that the system admits a global bounded weak solution under the same assumptions. Copyright © 2016 John Wiley & Sons, Ltd.