Global boundedness for a $ \mathit{\boldsymbol{N}} $-dimensional two species cancer invasion haptotaxis model with tissue remodeling

This paper is concerned with the two species cancer invasion haptotaxis model with tissue remodeling\begin{document}$ \begin{equation} \begin{cases} c_{1t} = \Delta c_1-\chi_1\nabla\cdot(c_1\nabla v)-\mu_{\rm EMT}c_1+\mu_1c_1(r_1-c_1^\kappa-c_2-v),\\ c_{2t} = \Delta c_2-\chi_2\nabla\cdot(c_2\nabla v)+\mu_{\rm EMT}c_1+\mu_2c_2(r_2-c_1-c_2^\kappa-v),\\ \tau m_t = \Delta m+c_1+c_2-m,\\ v_t = -mv+\eta v(1-c_1-c_2-v) \end{cases}\nonumber \end{equation} $\end{document}in a bounded and smooth domain \begin{document}$ \Omega\subset\mathbb{R}^N\;(N\geq1) $\end{document} with zero-flux boundary conditions for \begin{document}$ c_1,c_2 $\end{document} and \begin{document}$ m $\end{document} , where \begin{document}$ \chi_i,\mu_i,r_i>0\;(i = 1,2) $\end{document} , \begin{document}$ \eta>0 $\end{document} , \begin{document}$ \kappa\geq1 $\end{document} , \begin{document}$ \tau\in\{0,1\} $\end{document} , and \begin{document}$ \mu_{\rm EMT} = \mu_{ \rm EMT}\left(c_1,c_2,m,v\right):[0,\infty)^4\rightarrow [0,\infty) $\end{document} is the epithelial-mesenchymal transition rate function such that \begin{document}$ \mu_{\rm EMT}\leq\mu_M $\end{document} with some constant \begin{document}$ \mu_M>0 $\end{document} . When \begin{document}$ \kappa = 1 $\end{document} and \begin{document}$ N = 3 $\end{document} , by rasing the coupled a priori estimates of \begin{document}$ c_1 $\end{document} and \begin{document}$ c_2 $\end{document} in the following way \begin{document}$ L^1(\Omega)\rightarrow L^2(\Omega)\rightarrow L^p(\Omega)\rightarrow L^\infty(\Omega) $\end{document} with any \begin{document}$ p>2 $\end{document} , it is shown that for some appropriately regular and small initial data, the associated initial-boundary value problem possesses a unique globally bounded classical solution for suitably small \begin{document}$ r_i $\end{document} and \begin{document}$ \mu_M $\end{document} . When \begin{document}$ \kappa>1 $\end{document} and \begin{document}$ N\geq1 $\end{document} , by rasing the coupled a priori estimates of \begin{document}$ c_1 $\end{document} and \begin{document}$ c_2 $\end{document} from \begin{document}$ L^1(\Omega) $\end{document} to \begin{document}$ L^p(\Omega) $\end{document} with any \begin{document}$ p>1 $\end{document} , then to \begin{document}$ L^\infty(\Omega) $\end{document} , it is proved that for any reasonably regular initial data, the corresponding initial-boundary value problem admits a unique globally bounded classical solution for arbitrary \begin{document}$ r_i $\end{document} and \begin{document}$ \mu_M $\end{document} . The result for \begin{document}$ \kappa = 1 $\end{document} complements previously known one, and the result for \begin{document}$ \kappa>1 $\end{document} is new.