Radial solutions to a chemotaxis-consumption model involving prescribed signal concentrations on the boundary

TL;DR

This paper investigates radially symmetric solutions to a chemotaxis-consumption model with prescribed boundary signals, establishing global existence of solutions in 2D and higher dimensions, and analyzing stationary problem solvability.

Contribution

It provides new results on global bounded solutions for a chemotaxis system with boundary conditions, including weak solutions in 3-5 dimensions and stationary problem uniqueness.

Findings

Global bounded solutions exist in 2D for radially symmetric data.

Weak solutions are constructed for dimensions 3, 4, and 5.

Unique classical solutions are established for the stationary problem.

Abstract

The chemotaxis system \begin{align*} u_t &= \Delta u - \nabla \cdot (u\nabla v), \\ v_t &= \Delta v - uv, \end{align*} is considered under the boundary conditions $\frac{\partial u}{\partial\nu}- u\frac{\partial v}{\partial\nu}=0$ and $v=v_\star$ on $\partial\Omega$, where $\Omega\subset\mathbb{R}^n$ is a ball and $v_\star$ is a given positive constant. In the setting of radially symmetric and suitably regular initial data, a result on global existence of bounded classical solutions is derived in the case $n=2$, while global weak solutions are constructed when $n\in \{3,4,5\}$. This is achieved by analyzing an energy-type inequality reminiscent of global structures previously observed in related homogeneous Neumann problems. Ill-signed boundary integrals newly appearing therein are controlled by means of spatially localized smoothing arguments revealing higher order regularity features outside the spatial origin. Additionally, unique classical solvability in the corresponding stationary problem is asserted, even in nonradial frameworks.