A quasilinear chemotaxis-haptotaxis system: Existence and blow-up results

We consider the following chemotaxis-haptotaxis system: {u<inf>t</inf>=∇⋅(D(u)∇u)−χ∇⋅(S(u)∇v)−ξ∇⋅(u∇w),x∈Ω, t>0,v<inf>t</inf>=Δv−v+u,x∈Ω, t>0,w<inf>t</inf>=−vw,x∈Ω, t>0, under homogeneous Neumann boundary conditions in a bounded domain Ω⊂Rⁿ,n≥3 with smooth boundary. It is proved that for [Formula presented]≤A(s+1)^α for α<[Formula presented] and under suitable growth conditions on D, there exists a uniform-in-time bounded classical solution. Also, we prove that for radial domains, when the opposite inequality holds, the corresponding solutions blow-up in finite or infinite-time. We also provide the global-in-time existence and boundedness of solutions to the above system with small initial data when D(s)=1,S(s)=s.