Global classical solutions to the Cauchy problem of conservation laws with degenerate diffusion

In this paper we prove the existence of the unique global classical solution with small initial data to the Cauchy problem of a scalar conservation law with degenerate diffusion by establishing the uniform a priori   decay estimates of solutions. In order to compensate the degeneracy on the x1x1 direction by the diffusion on other directions, we introduce the frequency decomposition method and obtain the low frequency estimate and the high frequency estimate of the solution by the Green's function method and energy method respectively.