Global solution for a kinetic chemotaxis model with internal dynamics and its fast adaptation limit

TL;DR

This paper establishes the global existence of solutions for a nonlinear kinetic chemotaxis model with internal dynamics and analyzes its fast adaptation limit, revealing insights into chemotactic behavior and internal state concentration effects.

Contribution

It proves global existence of weak solutions using Schauder fixed point theorem and derives the model's fast adaptation limit, showing concentration effects and convergence to a Dirac mass.

Findings

Global weak solutions are established under general assumptions.

The fast adaptation limit results in solution concentration as a Dirac mass.

Convergence and compactness are rigorously demonstrated for the internal state and chemical potential.

Abstract

A nonlinear kinetic chemotaxis model with internal dynamics incorporating signal transduction and adaptation is considered. This paper is concerned with: (i) the global solution for this model, and, (ii) its fast adaptation limit to Othmer-Dunbar-Alt type model. This limit gives some insight to the molecular origin of the chemotaxis behaviour. First, by using the Schauder fixed point theorem, the global existence of weak solution is proved based on detailed a priori estimates, under some quite general assumptions on the model and the initial data. However, the Schauder fixed point theorem does not provide uniqueness. Therefore, additional analysis is required to be developed to obtain uniqueness. Next, the fast adaptation limit of this model is derived by extracting a weak convergence subsequence in measure space. For this limit, the first difficulty is to show the concentration effect on the internal state. When the small parameter {\epsilon}, the adaptation time scale, goes to zero, we prove that the solution converges to a Dirac mass in the internal state variable. Another difficulty is the strong compactness argument on the chemical potential, which is essential for passing the nonlinear kinetic equation to the weak limit.