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{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} $

in a bounded and smooth domain $ \Omega\subset\mathbb{R}^N\;(N\geq1) $ with zero-flux boundary conditions for $ c_1,c_2 $ and $ m $, where $ \chi_i,\mu_i,r_i>0\;(i = 1,2) $, $ \eta>0 $, $ \kappa\geq1 $, $ \tau\in\{0,1\} $, and $ \mu_{\rm EMT} = \mu_{ \rm EMT}\left(c_1,c_2,m,v\right):[0,\infty)^4\rightarrow [0,\infty) $ is the epithelial-mesenchymal transition rate function such that $ \mu_{\rm EMT}\leq\mu_M $ with some constant $ \mu_M>0 $. When $ \kappa = 1 $ and $ N = 3 $, by rasing the coupled a priori estimates of $ c_1 $ and $ c_2 $ in the following way $ L^1(\Omega)\rightarrow L^2(\Omega)\rightarrow L^p(\Omega)\rightarrow L^\infty(\Omega) $ with any $ p>2 $, 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 $ r_i $ and $ \mu_M $. When $ \kappa>1 $ and $ N\geq1 $, by rasing the coupled a priori estimates of $ c_1 $ and $ c_2 $ from $ L^1(\Omega) $ to $ L^p(\Omega) $ with any $ p>1 $, then to $ L^\infty(\Omega) $, 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 $ r_i $ and $ \mu_M $. The result for $ \kappa = 1 $ complements previously known one, and the result for $ \kappa>1 $ is new.