Pointwise estimates of solutions to conservation laws with nonlocal dissipation-type terms

We are concerned with the pointwise estimates of solutions to the scalar conservation law with a nonlocal dissipative term for arbitrary large initial data. Based on the Green's function method, time-frequency decomposition method as well as the classical energy estimates, pointwise estimates and the optimal decay rates are established in this paper. We emphasize that the decay rate is independent of the index s in the nonlocal dissipative term. This phenomenon is also coincident with the fact that the decay rate is determined by the low frequency part of the solution no matter the initial data is small or large.