Conservation laws with a degenerate source: Traveling waves, large-time behavior and zero relaxation limit

The large time behavior and zero reaction time limits of solutions of scalar hyperbolic conservation laws with bistable source terms whose zeroes are degenerate are studied. It is proved that as t→∞, solutions with periodic boundary condition tend to stable and unstable zeroes of the source terms. An example is given which shows that there exists initial data for which the solution converges pointwise to the unstable zero of the source term. Patterns of traveling waves of the systems and the conditions under which they exist are provided. It turns out that the structure of traveling waves and the long-time behavior are different from the case where the zeroes are non-degenerate. The recent results of Fan, Jin and Teng concerning the zero reaction time limit is generalized to the case of source terms with degenerate zeroes. The zero reaction time limits are quite different from the long time limit.