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 $\left\{\begin{aligned}\hfill & {u}_{t}={\Delta}u-\chi \mathbf{\nabla }\cdot \left(u\mathbf{\nabla }v\right)-\xi \mathbf{\nabla }\cdot \left(u\mathbf{\nabla }w\right)+\mu u\left(1-u\right)+\beta uv,\hfill \\ \hfill & {v}_{t}=D{\Delta}v+u-uv,\hfill \\ \hfill & {w}_{t}=-\delta vw+\eta w\left(1-w\right).\hfill \end{aligned}\right.$

We consider this problem in a bounded domain ${\Omega}\subset {\mathbb{R}}^{N}$ (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 $\left(1+\frac{\beta }{\mu },1,0\right)$ in the large time limit if δ > η; and strongly converges to $\left(1+\frac{\beta }{\mu },1,1-\frac{\delta }{\eta }\right)$ if δ < η. Unfortunately, for the case δ = η, 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 ${\Vert}\mathbf{\nabla }w{{\Vert}}_{{L}^{p}}$. 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.