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

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 quite general assumptions. However, the Schauder theorem does not provide uniqueness, so additional analysis is required to be developed for 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. Another difficulty is the strong compactness argument on the chemical potential, which is essential for passing the nonlinear kinetic equation to the weak limit.