A New Result for Global Solvability of a Two Species Cancer Invasion Haptotaxis Model with Tissue Remodeling

Source abstract

This paper deals with the two species cancer invasion haptotaxis model $c_{1t}=\Delta c_1-\chi_1\nabla\cdot(c_1\nabla v)-\mu_{EMT}c_1+\mu_1c_1(r_1-c_1-c_2-v);\;c_{2t}=\Delta c_2-\chi_2\nabla\cdot(c_2\nabla v)+\mu_{EMT}c_1+ \mu_2c_2(r_2-c_1-c_2-v);\;\tau m_t=\Delta m+c_1+c_2-m;\;v_t=-mv+\eta v(1-c_1-c_2-v),\;x\in\Omega,\;t>0$, in a bounded and smooth domain $\Omega\subset\mathbb{R}^N\;(N=2,3)$ with no-flux boundary conditions, where $\chi_1,\chi_2,r_1,r_2,\eta>0$, $\mu_1,\mu_2\geq0$, $\tau\in\{0,1\}$, and $\mu_{EMT}:=\mu_{EMT}(c_1,c_2,m,v)=f(v)m$ with $f\in C^1\left([0,\infty);[0,\infty)\right)$. Compared with a previous work [F. Dai and B. Liu, J. Math. Anal. Appl., 483 (2020), 123583], the novelties here are that the function $\mu_{EMT}(c_1,c_2,m,v)$ is not bounded, and that the logistic source in cell equations is possibly not taken into account. For any appropriately regular initial data, we prove the global existence and uniqueness of the classical solution to the corresponding two-dimensional system when $\mu_1\geq0,\mu_2=0$, and $\tau\in\{0,1\}$, and obtain the boundedness of this solution under an explicit smallness condition on $\eta$ when $\mu_1=\mu_2=0$ and $\tau=1$. Moreover, we establish the global existence, uniqueness, and boundedness of the classical solution to the associated three-dimensional system under some explicit smallness conditions on $r_1,r_2$, and initial dada when $\mu_1,\mu_2>0$, and $\tau\in\{0,1\}$. These results significantly improve and extend the result obtained for previously known ones.