# Renormalized solutions to a chemotaxis system with consumption of chemoattractant

## TL;DR

This paper establishes the existence of at least one global renormalized solution for a high-dimensional chemotaxis system with chemoattractant consumption, under specific initial data conditions and Neumann boundary conditions.

## Contribution

It proves the existence of global renormalized solutions for a chemotaxis system in dimensions four and higher with particular initial data requirements.

## Key findings

- Existence of global renormalized solutions in dimensions n≥4.
- Solutions exist under initial data in specific function spaces.
- The model accounts for chemoattractant consumption and boundary conditions.

## Abstract

This paper investigates a high-dimensional chemotaxis system with consumption of chemoattractant \begin{eqnarray*} \left\{\begin{array}{l} u_t=\Delta u-\nabla\cdot(u\nabla v), v_t=\Delta v-uv, \end{array}\right. \end{eqnarray*} under homogeneous boundary conditions of Neumann type, in a bounded convex domain $\Omega\subset\mathbb{R}^n~(n\geq4)$ 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.