Global existence and optimal decay rates of solutions to conservation laws with diffusion‐type terms

Abstract

We consider the Cauchy problem on nonlinear scalar conservation laws with a diffusion-type source term. Based on a low-frequency and high-frequency decomposition, Green's function method and the classical energy method, we not only obtain L 2 time-decay estimates but also establish the global existence of solutions to Cauchy problem when the initial data u 0 .x/ satisfies the smallness condition on kr u 0 k H s , but not on ku 0 k L 2 . Furthermore, by taking a time-frequency decomposition, we obtain the optimal decay estimates of solutions.