Summary: We model traffic flow with a time-dependent fundamental diagram. A time-dependent fundamental diagram arises naturally from various factors such as weather conditions, traffic jam or modern traffic congestion managements, etc. The model is derived from a car-following model which takes into account the situation changes over the time elapsed time. It is a system of non-concave hyperbolic conservation laws with time-dependent flux and the sources.
The global existence and uniqueness of the solution to the Cauchy problem is established under the condition that the variation in time of the fundamental diagram is bounded. The zero relaxation limit of the solutions is found to be the unique entropy solution of the equilibrium equation.