# American Institute of Mathematical Sciences

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June  2009, 24(2): 511-521. doi: 10.3934/dcds.2009.24.511

## Critical thresholds in a relaxation system with resonance of characteristic speeds

 1 Department of Mathematics, University of Iowa, 14 MacLean Hall, Iowa City, IA 52242-1419 2 Department of Mathematics, Iowa State University, Ames, IA 50011, United States

Received  November 2007 Revised  August 2008 Published  March 2009

We study critical threshold phenomena in a dynamic continuum traffic flow model known as the Payne and Whitham (PW) model. This model is a quasi-linear hyperbolic relaxation system, and when equilibrium velocity is specifically associated with pressure, the equilibrium characteristic speed resonates with one characteristic speed of the full relaxation system. For a scenario of physical interest we identify a lower threshold for finite time singularity in solutions and an upper threshold for the global existence of the smooth solution. The set of initial data leading to global smooth solutions is large, in particular allowing initial velocity of negative slope.
Citation: Tong Li, Hailiang Liu. Critical thresholds in a relaxation system with resonance of characteristic speeds. Discrete & Continuous Dynamical Systems, 2009, 24 (2) : 511-521. doi: 10.3934/dcds.2009.24.511
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