Stationary states of a chemotaxis consumption system with singular sensitivity and inhomogeneous boundary conditions

TL;DR

This paper proves the unique existence of stationary solutions for a chemotaxis system with singular sensitivity and inhomogeneous boundary conditions in bounded domains, advancing understanding of chemotactic behavior in mathematical biology.

Contribution

It establishes the first rigorous proof of unique stationary states for a chemotaxis model with singular sensitivity and specific boundary conditions.

Findings

Unique solvability of the stationary system in 2D and higher dimensions.

Explicit conditions on total mass for existence of solutions.

Mathematical framework applicable to biological chemotaxis models.

Abstract

For given total mass $m>0$ we show unique solvability of the stationary chemotaxis-consumption model \[ \begin{cases} 0= \Delta u - \chi \nabla \cdot (\frac{u}{v} \nabla v) \\ 0= \Delta v - uv \\ \int_\Omega u = m \end{cases} \] under no-flux-Dirichlet boundary conditions in bounded smooth domains $\Omega\subset \mathbb{R}^2$ and $\Omega=B_R(0)\subset \mathbb{R}^d$, $d\ge 3$.