On a Nonlinear Hyperbolic–Elliptic System Modeling Chemotaxis

In this paper, we study a hyperbolic-elliptic system with L1-initial data, which arises as a mathematical model for chemotaxis. In particular, after introducing the notion of entropy solutions in this setting, we prove results concerning the existence of solutions-either global or local in time-depending on how small the L1-norm of the initial data is. Furthermore, we prove the uniqueness of the entropy solutions by adapting the standard doubling-variable method for hyperbolic conservation laws to the current framework.