Global Solutions of Nonlinear Transport Equations for Chemosensitive Movement

A widespread phenomenon in moving microorganisms and cells is their ability to reorient themselves depending on changes of concentrations of certain chemical signals. In this paper we discuss kinetic models for chemosensitive movement, which also takes into account evaluations of gradient fields of chemical stimuli which subsequently influence the motion of the respective microbiological species. The basic type of model was discussed by Alt [J. Math. Biol., 9 (1980), pp. 147–177], [J. Reine Angew. Math., 322 (1981), pp. 15–41] and by Othmer, Dunbar, and Alt [J. Math. Biol., 26 (1988), pp. 263–298]. Chalub et al. rigorously proved that, in three dimensions, these kinds of kinetic models lead to the classical Keller–Segel model as its drift-diffusion limit when the equation for the chemo-attractant is of elliptic type [Monatsh. Math., 142 (2004), pp. 123–141], [On the Derivation of Drift-Diffusion Model for Chemotaxis from Kinetic Equations, ANUM preprint 14/02, Vienna Technical University, 2002]. In [H. Hwang, K. Kang, and A. Stevens, Drift-diffusion limits of kinetic models for chemotaxis: A generalization, Discrete Contin. Dyn. Syst. Ser. B., to appear] it was proved that the macroscopic diffusion limit exists in both two and three dimensions also when the equation of the chemo-attractant is of parabolic type. So far in the rigorous derivations, only the density of the chemo-attractant was supposed to influence the motion of the chemosensitive species. Here we show that in the macroscopic limit some types of evaluations of gradient fields of the chemical stimulus result in a change of the classical parabolic Keller–Segel model for chemotaxis. Under suitable structure conditions, global solutions for the kinetic models can be shown.