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.