Renormalized solutions to a chemotaxis system with consumption of chemoattractant

Abstract

This paper investigates a high-dimensional chemotaxis system with consumption of chemoattractant \begin{eqnarray*} \left\{\begin{array}{l} u_t=Δu-\nabla\cdot(u\nabla v), v_t=Δv-uv, \end{array}\right. \end{eqnarray*} under homogeneous boundary conditions of Neumann type, in a bounded convex domain $\Omega \subset {\mathbb{R}}^n(n\ge 4)$ with smooth boundary. It is proved that if initial data satisfy $u_0\in C^0(\overline{\Omega })$ and $v_0\in W^{1,q}(\Omega )$ for some $q>n$, the model possesses at least one global renormalized solution.

12 pages