Boundedness of solutions to a quasilinear chemotaxis–haptotaxis model

We study global solutions of a class of chemotaxis–haptotaxis systems generalizing the prototype {ut=∇⋅((u+1)m−1∇u)−∇⋅(u(u+1)q−1∇v)−∇⋅(u(u+1)p−1∇w)+H(u,w),0=Δv−v+u,wt=−vw, in a bounded domain Ω⊂RN(N≥1)Ω⊂RN(N≥1) with smooth boundary, H(u,w):=u(1−ur−1−w)H(u,w):=u(1−ur−1−w), with parameters m≥1,r>1m≥1,r>1 and positive constants p,qp,q. It is shown that either max{q+1,p,2p−m}0b>0 is large enough, then for any sufficiently smooth initial data there exists a classical solution which is global in time and bounded. The results of this paper improve the results of Tao and Winkler (2014) [46,51].