Boundedness of solutions to a quasilinear chemotaxis-haptotaxis model

We study global solutions of a class of chemotaxis-haptotaxis systems generalizing the prototype {u_t=∇·((u+1)^m-1∇u)-∇·(u(u+1)^q-1∇v)-∇·(u(u+1)^p-1∇w)+H(u,w),0=Δv-v+u, w_t=-vw, in a bounded domain Ω ⊂ ℝ^N(N≥1) with smooth boundary, H(u,w):=u(1-u^r-1-w), with parameters m≥1,r>1 and positive constants p,q. It is shown that either max{q+1,p,2p-m}<max{m+2/N,r} or max{q+1,p,2p-m}=r and b>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].