A note for global existence of a two-dimensional chemotaxis-haptotaxis model with remodeling of non-diffusible attractant

In this paper, we study the following the coupled chemotaxis--haptotaxis model with remodeling of non-diffusible attractant $$ \left\{\begin{array}{ll} u_t = \Delta u-\chi\nabla\cdot(u\nabla v)- \xi\nabla\cdot(u\nabla w)+\mu u(1- u-w), \displaystyle{v_t=\Delta v- v +u},\quad \displaystyle{w_t=- vw+\eta w(1-u-w)},\quad \end{array}\right. $$ in a bounded smooth domain $\mathbb{R}^2$ with zero-flux boundary conditions, where $\chi$, $\xi$ and $\eta$ are positive parameters. Under appropriate regularity assumptions on the initial data $(u_0, v_0, w_0)$, by develops some $L^p$-estimate techniques, we prove the global existence and uniqueness of classical solutions when $\mu>0$ (where $\mu$ is the logistic growth rate of cancer cells). Here we use an approach based on maximal Sobolev regularity and the variation-of-constants formula remove the restrictions $\mu$ is sufficiently large, which required in \cite{PangPang1}.