Asymptotic behavior of solutions to conservation laws with diffusion-type terms of regularity-gain and regularity-loss

Abstract

In this paper, we study asymptotic behaviors of solutions to the Cauchy problem of nonlinear conservation laws with a diffusion-type source term related to an index s∈R. For s≤1 and s>1, the diffusion-type term takes on a characteristic of regularity-gain and regularity-loss on the high frequency domain, respectively. By combining the Green function method with the energy method, we overcome the weakly dissipative structure of the equation for the case of s>1 and obtain the global existence and optimal Lp-norm time-decay rates of solutions. In the case of regularity-gain, pointwise estimates of solutions are shown by using the refined analysis on the Green function.