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

Abstract

<p style='text-indent:20px;'>This paper is concerned with the two species cancer invasion haptotaxis model with tissue remodeling</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \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} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>in a bounded and smooth domain <inline-formula><tex-math id="M2">\begin{document}$ \Omega\subset\mathbb{R}^N\;(N\geq1) $\end{document}</tex-math></inline-formula> with zero-flux boundary conditions for <inline-formula><tex-math id="M3">\begin{document}$ c_1,c_2 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M4">\begin{document}$ m $\end{document}</tex-math></inline-formula>, where <inline-formula><tex-math id="M5">\begin{document}$ \chi_i,\mu_i,r_i&gt;0\;(i = 1,2) $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M6">\begin{document}$ \eta&gt;0 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M7">\begin{document}$ \kappa\geq1 $\end{document}</tex-math></inline-formula>, <inline-formula><tex-math id="M8">\begin{document}$ \tau\in\{0,1\} $\end{document}</tex-math></inline-formula>, and <inline-formula><tex-math id="M9">\begin{document}$ \mu_{\rm EMT} = \mu_{ \rm EMT}\left(c_1,c_2,m,v\right):[0,\infty)^4\rightarrow [0,\infty) $\end{document}</tex-math></inline-formula> is the epithelial-mesenchymal transition rate function such that <inline-formula><tex-math id="M10">\begin{document}$ \mu_{\rm EMT}\leq\mu_M $\end{document}</tex-math></inline-formula> with some constant <inline-formula><tex-math id="M11">\begin{document}$ \mu_M&gt;0 $\end{document}</tex-math></inline-formula>. When <inline-formula><tex-math id="M12">\begin{document}$ \kappa = 1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M13">\begin{document}$ N = 3 $\end{document}</tex-math></inline-formula>, by rasing the coupled a priori estimates of <inline-formula><tex-math id="M14">\begin{document}$ c_1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M15">\begin{document}$ c_2 $\end{document}</tex-math></inline-formula> in the following way <inline-formula><tex-math id="M16">\begin{document}$ L^1(\Omega)\rightarrow L^2(\Omega)\rightarrow L^p(\Omega)\rightarrow L^\infty(\Omega) $\end{document}</tex-math></inline-formula> with any <inline-formula><tex-math id="M17">\begin{document}$ p&gt;2 $\end{document}</tex-math></inline-formula>, it is shown that for some appropriately regular and small initial data, the associated initial-boundary value problem possesses a unique globally bounded classical solution for suitably small <inline-formula><tex-math id="M18">\begin{document}$ r_i $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M19">\begin{document}$ \mu_M $\end{document}</tex-math></inline-formula>. When <inline-formula><tex-math id="M20">\begin{document}$ \kappa&gt;1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M21">\begin{document}$ N\geq1 $\end{document}</tex-math></inline-formula>, by rasing the coupled a priori estimates of <inline-formula><tex-math id="M22">\begin{document}$ c_1 $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M23">\begin{document}$ c_2 $\end{document}</tex-math></inline-formula> from <inline-formula><tex-math id="M24">\begin{document}$ L^1(\Omega) $\end{document}</tex-math></inline-formula> to <inline-formula><tex-math id="M25">\begin{document}$ L^p(\Omega) $\end{document}</tex-math></inline-formula> with any <inline-formula><tex-math id="M26">\begin{document}$ p&gt;1 $\end{document}</tex-math></inline-formula>, then to <inline-formula><tex-math id="M27">\begin{document}$ L^\infty(\Omega) $\end{document}</tex-math></inline-formula>, it is proved that for any reasonably regular initial data, the corresponding initial-boundary value problem admits a unique globally bounded classical solution for arbitrary <inline-formula><tex-math id="M28">\begin{document}$ r_i $\end{document}</tex-math></inline-formula> and <inline-formula><tex-math id="M29">\begin{document}$ \mu_M $\end{document}</tex-math></inline-formula>. The result for <inline-formula><tex-math id="M30">\begin{document}$ \kappa = 1 $\end{document}</tex-math></inline-formula> complements previously known one, and the result for <inline-formula><tex-math id="M31">\begin{document}$ \kappa&gt;1 $\end{document}</tex-math></inline-formula> is new.</p>