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

We consider the following chemotaxis-haptotaxis system: {u(t) = del center dot (D(u)del u) - chi del center dot (S(u)del v) - xi del center dot (u del w), x is an element of Omega, t > 0, v(t) = Delta v - v + u, x is an element of Omega, t > 0, w(t) = -vw, x is an element of Omega, t > 0, under homogeneous Neumann boundary conditions in a bounded domain Omega subset of R-n, n >= 3with smooth boundary. It is proved that for S(s)/D(s) <= A(s+ 1)(alpha) for alpha < 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. (c) 2024 Elsevier Inc. All rights reserved.