Boundedness of solution of a parabolic--ODE--parabolic chemotaxis--haptotaxis model with (generalized) logistic source

In this paper, we study the following chemotaxis–haptotaxis system with (generalized) logistic source $$\{\begin{array}{l}{u}_t=\mathrm{\Delta }u-\chi \mathrm{\nabla }\cdot (u\mathrm{\nabla }v)-\xi \mathrm{\nabla }\cdot (u\mathrm{\nabla }w)+u(a-\mu u^{r-1}-w),{\displaystyle v_t=\mathrm{\Delta }v-v+u,}\\ {\displaystyle w_t=-vw,}\\ {\displaystyle \frac{\mathrm{\partial }u}{\mathrm{\partial }\nu }=\frac{\mathrm{\partial }v}{\mathrm{\partial }\nu }=\frac{\mathrm{\partial }w}{\mathrm{\partial }\nu }=0,x\in \mathrm{\partial }\mathrm{\Omega },t>0,}\\ {\displaystyle u(x,0)=u_0(x),v(x,0)=v_0(x),w(x,0)=w_0(x),x\in \mathrm{\Omega },}\end{array}\text{eqno}(0.1)$$ %under homogeneous Neumann boundary conditions in a smooth bounded domain ${\mathbb{R}}^N(N\ge 1)$, with parameter $r>1$. the parameters $a\in \mathbb{R},\mu >0,\chi >0$. It is shown that when $r>2$, or \begin{equation*} \mu>\mu^{}=\begin{array}{ll} \frac{(N-2){+}}{N}(\chi+C{\beta}) C^{\frac{1}{\frac{N}{2}+1}}_{\frac{N}{2}+1},~~~\mbox{if}~~r=2, \end{array} \end{equation} % $\mu>\frac{(N-2){+}}{N}\chi C^{\frac{1}{\frac{N}{2}+1}}{\frac{N}{2}+1}$, the considered problem possesses a global classical solution which is bounded, where $C_{\frac{N}{2}+1}^{\frac{1}{\frac{N}{2}+1}}$ is a positive constant which is corresponding to the maximal sobolev regularity. Here $C_{\beta }$ is a positive constant which depends on $\xi$, $\mathrm{\parallel }{u}_0{\mathrm{\parallel }}_{C(\bar{\mathrm{\Omega }})},\mathrm{\parallel }{v}_0{\mathrm{\parallel }}_{W^{1,\mathrm{\infty }}(\mathrm{\Omega })}$ and $\mathrm{\parallel }{w}_0{\mathrm{\parallel }}_{L^{\mathrm{\infty }}(\mathrm{\Omega })}$. This result improves or extends previous results of several authors.