Renormalized solutions to a chemotaxis system with consumption of chemoattractant

Abstract

This article concerns the high-dimensional chemotaxis system with consumption of chemoattractant ut = ∆u - ∇ ∙ (u∇v), vt = ∆v - uv, under homogeneous boundary conditions of Neumann type, in a bounded domain Ω ⊂ ℝn (n ≥ 4) with smooth boundary. We prove that if the initial data satisfy u0 ∈ C0(Ω̅) and v0 ∈ W1,q(Ω) for some q > n, this model possesses at least one global renormalized solution.