Time-asymptotic behaviour of a solution of the Cauchy initial-value problem for a conservation law with non-linear divergent viscosity

We study the time-asymptotic behaviour of a solution of the Cauchy initial-value problem for a conservation law with non-linear divergent viscosity. We shall prove that when a bounded measurable initial function has limits at $\pm\infty$, a solution of the Cauchy initial-value problem converges uniformly to a system of waves consisting of travelling waves and rarefaction waves, where the phase shifts of the travelling waves are allowed to depend on time. The rate of convergence is estimated under additional conditions on the initial function.