Global solvability and stabilization to a cancer invasion model with remodelling of ECM

In this paper, we deal with the Chaplain-Lolas's model of cancer invasion with tissue remodelling (u(t) = Delta u - chi del center dot (u del v) - xi del center dot (u del w) + mu u(1 - u) + beta,uv, v(t) = D Delta v + u - uv, w(t) = - delta vw + eta w(1 - w). We consider this problem in a bounded domain Omega subset of R-N (N = 2, 3) with zero-flux boundary conditions. We first establish the global existence and uniform boundedness of solutions. Subsequently, we also consider the large time behaviour of solutions, and show that the global classical solution (u, v, w) strongly converges to the semi-trivial steady state (1 + beta/mu, 1, 0) in the large time limit if delta > eta; and strongly converges to (1 + beta/mu, 1, 1 - delta/eta) if delta < eta. Unfortunately, for the case delta = eta, we only prove that (v, w) -> (1, 0), and it is hard to obtain the large time limit of u due to lack of uniform boundedness of parallel to del w parallel to(Lp). It is worth noting that the large time behaviour of solutions for the chemotaxis-haptotaxis model with tissue remodelling has never been touched before, this paper is the first attempt. At last, taking advantage of the large time behaviour of solutions, we also establish the uniform boundedness of solutions in the classical sense.