Asymptotics of a chemotaxis-consumption-growth model with nonzero Dirichlet conditions

TL;DR

This paper analyzes the long-term behavior of a chemotaxis-consumption model with logistic growth, proving global bounded solutions in 2D and convergence to steady state under certain boundary conditions.

Contribution

It establishes existence, boundedness, and convergence results for a chemotaxis model with boundary chemoattractant concentration, extending understanding of its asymptotic behavior.

Findings

Existence of unique global classical solutions in 2D.

Solutions are uniformly bounded over time.

Convergence to positive steady state when boundary chemoattractant is low.

Abstract

This paper concerns the asymptotics of certain parabolic-elliptic chemotaxis-consumption systems with logistic growth and constant concentration of chemoattractant on the boundary. First we prove that in two dimensional bounded domains there exists a unique global classical solution which is uniformly bounded in time, and then we show that if the concentration of chemoattractant on the boundary is sufficiently low then the solution converges to the positive steady state as time goes to infinity.