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 $$\{\begin{array}{l}{u}_t=\mathrm{\Delta }u-\chi \mathrm{\nabla }\cdot (u\mathrm{\nabla }v)-\xi \mathrm{\nabla }\cdot (u\mathrm{\nabla }w)+\mu u(1-u-w),{\displaystyle v_t=\mathrm{\Delta }v-v+u,{\displaystyle w_t=-vw+\eta w(1-u-w),}}\end{array}$$ 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 $(u_0, v_0, w_0)$,bydevelopssome$L^p$-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}.