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We show that in one space dimension, a linearly degenerate hyperbolic systemof rich type admits exact traveling wave solutions if the initial data areRiemann type outside of a space interval. In a particular case of the systemincluding physical models, we prove the convergence of entropy solutions totraveling waves in the $L^1$ norm as the time goes to infinity. The travelingwaves are determined explicitly in terms of the initial data and the system. Wealso obtain the stability of entropy solutions in $L^1$.

Long-time behaviors and stability of entropy solutions for linearly degenerate hyperbolic systems of rich type