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

We study a scalar hyperbolic conservation law with a bistable source term whose zeroes are degenerate. It turns out that the structure of the set of traveling waves as well as the long-time behavior is dierent from the case where the zeroes are non-degenerate. In particular, for x 2 S¹, there exist initial data for which the solution converges pointwise to the unstable zero of the source term. We also generalize recent results of Fan, Jin and Teng concerning the zero reaction time limit to the case of source terms with degenerate zeroes. This shows that in general the long time limit and the zero reaction time limit cannot be interchanged.