Uniform in time solutions for a chemotaxis with potential consumption model

TL;DR

This paper proves the existence of uniform in time solutions for a chemotaxis model with nonlinear potential consumption, demonstrating different properties in 1D, 2D, and 3D domains under minimal boundary regularity.

Contribution

It introduces a novel chemotaxis model with a nonlinear potential consumption term and establishes existence, uniqueness, and regularity results with minimal boundary regularity assumptions.

Findings

Existence of uniform in time weak solutions in 3D domains.

Uniqueness and regularity in 2D and 1D domains.

Minimal boundary regularity assumptions are sufficient for the results.

Abstract

In this work we investigate the following chemo-attraction with consumption model in bounded domains of \, $\mathbb{R}^N$ ($N=1,2,3$): $$ \partial_t u - \Delta u = - \nabla \cdot (u \nabla v), \quad \partial_t v - \Delta v = - u^s v $$ where $s\ge 1$, endowed with isolated boundary conditions and initial conditions for $(u,v)$. The main novelty in the model is the nonlinear potential consumption term $u^sv$. Through the convergence of solutions of an adequate truncated model, two main results are established; existence of uniform in time weak solutions in $3D$ domains, and uniqueness and regularity in $2D$ (or $1D$) domains. Both results are proved imposing minimal regularity assumptions on the boundary of the domain.