Local Well-posedness of Kinetic Chemotaxis Models

We present a general functional analytic setting in which the Cauchy problem for mild solutions of kinetic chemotaxis models is well-posed, locally in time, in general physical dimensions. The models consist of a hyperbolic transport equation that is non-linearly and non-locally coupled to a reaction-diffusion system through kernel operators. Three examples are elaborated throughout the paper in which the latter system is (1) a single linear equation, (2) a FitzHugh-Nagumo system and (3) a piecewise linear approximation thereof. Finally we present a limit argument to obtain results on solutions in L 1 ∩ L ∞.