Large Time Behavior and Diffusion Limit for a System of Balance Laws From Chemotaxis in Multi-dimensions

TL;DR

This paper analyzes the long-term behavior and diffusion limits of a multi-dimensional chemotaxis system modeled by balance laws, establishing global solutions and their convergence to equilibrium and simplified models.

Contribution

It proves global well-posedness for small initial energy and demonstrates convergence to equilibrium and diffusion limits in a multi-dimensional chemotaxis system.

Findings

Global classical solutions exist under small initial energy.

Solutions converge to constant equilibrium states over time.

Solutions approach partially dissipative models as chemical diffusion vanishes.

Abstract

We consider the Cauchy problem for a system of balance laws derived from a chemotaxis model with singular sensitivity in multiple space dimensions. Utilizing energy methods, we first prove the global well-posedness of classical solutions to the Cauchy problem when only the energy of the first order spatial derivatives of the initial data is sufficiently small, and the solutions are shown to converge to the prescribed constant equilibrium states as time goes to infinity. Then we prove that the solutions of the fully dissipative model converge to those of the corresponding partially dissipative model when the chemical diffusion coefficient tends to zero.