Boundedness in a two-dimensional chemotaxis-haptotaxis system

Abstract

This work studies the chemotaxis-haptotaxis system $$\{\begin{array}{ll}{u}_t=\Delta u-\chi \nabla \cdot (u\nabla v)-\xi \nabla \cdot (u\nabla w)+\mu u(1-u-w),& x\in \Omega ,t>0,\\ v_t=\Delta v-v+u,& x\in \Omega ,t>0,\\ w_t=-vw,& x\in \Omega ,t>0,\end{array}$$ in a bounded smooth domain $\Omega \subset {\mathbb{R}}^2$ with zero-flux boundary conditions, where the parameters $\chi ,\xi$ and $\mu$ are assumed to be positive. It is shown that under appropriate regularity assumption on the initial data $(u_0,v_0,w_0)$, the corresponding initial-boundary problem possesses a unique classical solution which is global in time and bounded. In addition to coupled estimate techniques, a novel ingredient in the proof is to establish a one-sided pointwise estimate, which connects $\Delta w$ to $v$ and thereby enables us to derive useful energy-type inequalities that bypass $w$. However, we note that the approach developed in this paper seems to be confined to the two-dimensional setting.

17 pages