Boundedness in a two-dimensional two-species cancer invasion haptotaxis model without cell proliferation

Abstract

This paper reconsiders the two-species cancer invasion haptotaxis model without cell proliferation

$$\begin{aligned} {\left\{ \begin{array}{ll} c_{1t}=\Delta c_1-\chi _1\nabla \cdot (c_1\nabla v)-f(v)mc_1, \\ c_{2t}=\Delta c_2-\chi _2\nabla \cdot (c_2\nabla v)+f(v)mc_1, \\ \tau m_t=\Delta m+c_1+c_2-m, \\ v_t=-mv+\eta v(1-\alpha _1c_1-\alpha _2c_2-v) \end{array}\right. }\quad (\star ) \end{aligned}$$

in a bounded and smooth domain $\Omega \subset {\mathbb {R}}^2$ with homogeneous Neumann conditions, where $\chi _1,\chi _2,\eta >0$, $\tau \in \{0,1\}$, $f(v)\in C^1\left( [0,\infty );[0,\infty )\right)$ and $f(0)=0$. It is well known that the absence of logistic source aggravates mathematical difficulties, which are overcome by constructing suitable Lyapunov functional. When the remodeling of ECM includes a competition with cancer cells (i.e., $\alpha _1=\alpha _2=1$), we prove that the associated initial-boundary value problem of $(\star )$ with $\tau =0$ admits a globally bounded classical solution for suitably small $\eta$, which complements the boundedness result on the homogenous Neumann problem of $(\star )$ with $\tau =1$ obtained in Dai and Liu (SIAM J Math Anal 54:1–35, 2022). When the competition with cancer cells is taken no account in the re-establishment of ECM (i.e., $\alpha _1=\alpha _2=0$), we establish the global boundedness of classical solution to the corresponding initial-boundary value problem of $(\star )$ with $\tau \in \{0,1\}$ for arbitrarily large $\eta$, which is completely new. These results reveal the significant difference on the global boundedness of classical solution for the case whether or not the competition with cancer cells is contained in the remodeling of ECM.