Chemotaxis-consumption interaction: Solvability and asymptotics in general high-dimensional domains

TL;DR

This paper studies a chemotaxis-consumption model in high-dimensional, possibly non-convex domains, proving global existence, eventual smoothness, and stabilization of solutions over time.

Contribution

It establishes the global existence and long-term behavior of solutions to a general chemotaxis-consumption model in complex domains.

Findings

Global existence of weak solutions

Solutions become smooth over time

Solutions stabilize as time progresses

Abstract

The basic chemotaxis-consumption model \[ u_t = \Delta u - \nabla \cdot(u\nabla v),\qquad\qquad v_t = \Delta v - uv \] is considered in general, possibly non-convex bounded domains of arbitrary spatial dimension. Global existence of weak solutions is shown, along with eventual smoothness of solutions and their stabilization in the large time limit.