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

We consider the following chemotaxis-haptotaxis system: { u t = ∇ ⋅ (D (u) ∇ u) − χ ∇ ⋅ (S (u) ∇ v) − ξ ∇ ⋅ (u ∇ w) , x ∈ Ω , t > 0 , v t = Δ v − v + u , x ∈ Ω , t > 0 , w t = − v w , x ∈ Ω , t > 0 , under homogeneous Neumann boundary conditions in a bounded domain Ω ⊂ R n , n ≥ 3 with smooth boundary. It is proved that for S (s) D (s) ≤ A (s + 1) α for α < 2 n 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. [ABSTRACT FROM AUTHOR]