Abstract: We study the longtime behavior of spatially inhomogeneous scalar balance lawswith periodic initial data and a convex flux f.
Our main result states that for a large class of initial data the entropy solution will either converge uniformly to some steady state or to a discontinuous time-periodic solution.
This extends results of Lyberopoulos, Sinestrari and Fan & Hale obtained in the spatially homogeneous case.
The proof is based on the method of generalized characteristics together with ideas from dynamical systems theory.
A major difficulty consists of finding the periodic solutions which determine the asymptotic behavior. To this end we introduce a new tool, the Rankine–Hugoniot vector field, which describes the motion of a (hypothetical) shock with certain prescribed left and right states. We then show the existence of periodic solutions of the Rankine–Hugoniot vector field and prove that the actual shock curves converge to these periodic solutions.
Conservation Laws Preprint Server <email@example.com> 2004-10-08 12:49:17 UTC