Formation of delta shock waves in the limit of Riemann solutions to the Aw–Rascle traffic model with a damping term

In this paper, we consider the Riemann problem of the Aw–Rascle traffic model with a damping term and the formation of delta shock waves in the limit of the Riemann solutions as $\gamma \to 1$. By introducing a new variable and employing generalized characteristic analysis methods, we construct solutions to the Riemann problem of the inhomogeneous Aw–Rascle traffic model. Specially, for the case $0<u_{-}<u_{+}$, we prove the existence of a critical value ${\overline{\gamma }}_0$ for $\gamma$ such that when $0<\gamma <{\overline{\gamma }}_0$, the Riemann solutions contain no vacuum states; otherwise, a vacuum state emerges. Furthermore, we demonstrate that as $\gamma \to 1$, the limit of the Riemann solutions with vacuum states aligns with the Riemann solutions to the inhomogeneous transport model under the same initial conditions, while the limit of solutions with shock waves converges to a curved delta shock solution. Notably, the weights supported on the delta shock solution differ from the Riemann solutions to the inhomogeneous transport model due to the influence of the damping term.