Source-Solutions and Asymptotic Behavior in Conservation Laws.

Abstract We study the uniqueness of the solutions to the scalar conservation law u t + ϑ ( u ) x = 0 when the initial datum is a finite measure. The case of a Dirac mass is particularly emphasized: it is shown how it provides a description of the asymptotic behavior of the solutions initiated by an arbitrary integrable function. This behavior is proved to depend on one parameter in the case when ϑ is odd while it depends on two when ϑ is convex.