Improvement of conditions for boundedness in a chemotaxis consumption system with density-dependent motility

Abstract

This work considers the chemotaxis model with density-dependent motility ut=Δ(ϕ(v)u),x∈Ω,t>0,vt=Δv−uv,x∈Ω,t>0under homogeneous Neumann boundary conditions in a smooth bounded domain Ω⊂R2, with the nonnegative initial data (u0,v0)∈[W1,∞(Ω)]2. The motility function ϕ(v) satisfies ϕ(v)∈C3([0,∞)) with ϕ(v)>0 and ϕ′(v)<0 for all v≥0. For all suitably regular initial data, we prove that this system has a unique global bounded classical solution. The purpose of this work is to remove the smallness assumption on ‖v0‖L∞(Ω) in Li and Zhao (2021).