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**Stability of a traffic flow model with nonconvex relaxation.**
*(English)*
Zbl 1099.35156

Summary: This paper is concerned with the nonlinear stability of traveling wave solutions for a quasi-linear relaxation model with a nonconvex equilibrium flux. The study is motivated by and the results are applied to the well-known dynamic continuum traffic flow model, the Payne and Whitham (PW) model with a nonconcave fundamental diagram. The PW model is the first of its kind and it has been widely adopted by traffic engineers in the study of stability and instability phenomena of traffic flow. The traveling wave solutions are shown to be asymptotically stable under small disturbances and under the sub-characteristic condition using a weighted energy method. The analysis applies to both non-degenerate case and the degenerate case where the traveling wave has exponential decay rates at infinity and has an algebraic decay rate at infinity, respectively.

### MSC:

35Q80 | Applications of PDE in areas other than physics (MSC2000) |

35B35 | Stability in context of PDEs |

90B20 | Traffic problems in operations research |

35B40 | Asymptotic behavior of solutions to PDEs |

35L65 | Hyperbolic conservation laws |

76L05 | Shock waves and blast waves in fluid mechanics |