Global solvability of a chemotaxis-haptotaxis model in the whole 2-d space

Abstract

This paper investigates a two-dimensional chemotaxis-haptotaxis model $ \begin{eqnarray*} \left\{\begin{array}{lll} u_t = \Delta u-\chi\nabla\cdot(u\nabla v)-\xi\nabla\cdot(u\nabla w),&{} x\in\mathbb{R}^2,\ t>0,\\ v_t = \Delta v-v+u,&{}x\in\mathbb{R}^2,\ t>0,\\ w_t = -vw,&{}x\in\mathbb{R}^2,\ t>0, \end{array}\right. \end{eqnarray*} $ where $ \chi $ and $ \xi $ are positive parameters. It is proved that, for any suitable smooth initial data $ (u_0, v_0, w_0) $, this model admits a unique global strong solution if $ \left\|u_0\right\|_{L^1} < \frac{8 \pi}{\chi} $. Compared to the result by Calvez and Corrias (Calvez and Corrias, 2008 [1]), we can see that the haptotaxis effect is almost negligible in terms of global existence, which is consistent with the result of bounded domain (Jin and Xiang, 2021 [2]). Moreover, to the best of our knowledge, this is the first analytical work for the well-posedness of chemotaxis-haptotaxis system in the whole space.

Keywords

• chemotaxis
• haptotaxis
• global existence
• whole space
• cauchy problem