Nonlocal conservation laws with time delay

Keimer A, Pflug L (2019)

Publication Type: Journal article

Publication year: 2019


Book Volume: 26

Article Number: 54

Journal Issue: 6

DOI: 10.1007/s00030-019-0597-z


This study considers nonlocal conservation laws in which the velocity depends nonlocally on the solution not in real time but in a time-delayed manner. Nonlocal refers to the fact that the velocity of the conservation law depends on the solution integrated over a specific area in space. In every model modelling human’s behavior a time delay as reaction/response time is crucial. We distinguish a so called nonlocal classical delay model where only the velocity of the conservation law is delayed from a more realistic nonlocal delay model where also a shift backwards in space is considered. For both models we show existence and uniqueness of the solutions and study their analytical properties. We also present a direct application in traffic flow modelling. We also show that for delay approaching zero, the solutions of the considered delayed models converge to the solutions of the non-delayed models in the proper topology. Finally, a comprehensive numerical study illustrating exemplary the impacts of delay and nonlocality are presented and compared with nonlocal models without delay, as well as the corresponding local models.

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Keimer, A., & Pflug, L. (2019). Nonlocal conservation laws with time delay. Nodea-Nonlinear Differential Equations and Applications, 26(6).


Keimer, Alexander, and Lukas Pflug. "Nonlocal conservation laws with time delay." Nodea-Nonlinear Differential Equations and Applications 26.6 (2019).

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