Infinite Propagation Speed for a Two-Component Camassa-Holm Equation

The use of Partial Differential Equation models for studying traffic flows has a fairly long history. This paper deals with the Cauchy problem of a two-component Camassa-Holm equation. First, we prove that the solution ρ keeps the property of having compact support for any further time provided the initial data ρ_0 has compact support. While the initial data m_0 has compact support then the solution m will remain compactly supported, only if ρ is also initially compactly supported. Then, we get the infinite propagation speed in the sense that the solution u with compactly supported initial data does not have compact support any longer in its lifespan. Although the nontrivial solution u is no longer compactly supported, a detailed description about the profile of the solution u is shown as it evolves over time.