Global classical solution of chemotaxis-haptotaxis model with re-establishment mechanisms

This paper considers the following coupled chemotaxis-haptotaxis system with re-establishment mechanisms$$\begin{cases} u_t = \Delta u - \chi \nabla \cdot (u \nabla v) - \xi \nabla \cdot (u \nabla w) + f(u,w), \\ v_t = \Delta v - v + u, \\ w_t = -vw + \eta w(1-u-w) \end{cases} \quad (*)$$in a bounded domain $\Omega \subset \mathbb{R}^2$ corresponding to zero-flux boundary conditions, where $\chi$, $\xi$ and $\eta$ are positive parameters. This model describes the process of cancer cell invasion of surrounding healthy tissue. For system with re-establishment mechanisms (*), we are the first to establish the existence of solutions under the weaker conditions of $f$ (such as sub-logistic conditions), and innovatively derive the boundedness of the solutions, subsequently obtaining their long-time asymptotic behavior. Additionally, we are delighted to discover that by only adding constraints to the initial value of $u$, rather than to both $u$ and $v$ simultaneously, we can discern the asymptotic behavior of the solution, which also opens up a new perspective for the study of such problems.