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

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.