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

TL;DR

The article analyzes the prototypical chemotaxis-consumption system $u_t=\Delta u-\nabla\cdot(u\nabla v)$, $v_t=\Delta v-uv$ in general smooth bounded domains with Neumann boundary. It develops a novel refined energy framework based on $${\mathcal F}_\varepsilon$$ and $${\mathcal D}_\varepsilon$$, controlling the cross-diffusive term through the dissipation of $v$ and a boundary estimate, which yields dimension-independent $L^1$-type compactness. Using a regularized problem and Aubin–Lions compactness, the authors construct global weak solutions that become smooth for large times and converge to the homogeneous steady state $$(u,v)=(\mu,0)$$ with exponential rate, where $\mu=|\Omega|^{-1}\int_\Omega u_0$. This extends existing two-dimensional or convex-domain results to general high-dimensional geometries, advancing understanding of chemotaxis systems with signal consumption and their long-time behavior.

Abstract

The basic chemotaxis-consumption model \[ u_t = Δu - \nabla \cdot(u\nabla v),\qquad\qquad v_t = Δv - uv \] 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.

Theorems & Definitions (20)

• Definition 1.1
• Theorem 1.2
• Lemma 2.1
• Lemma 2.2
• Lemma 2.3
• Lemma 3.1
• Lemma 3.2
• Lemma 3.3
• Lemma 3.4
• Lemma 3.5