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This paper is concerned with the large time behaviors of the entropysolutions to one-dimensional scalar convex conservation laws, of which theinitial data are assumed to approach two arbitrary $ L^\infty $ periodicfunctions as $ x\rightarrow-\infty $ and $ x\rightarrow+\infty, $ respectively.We show that the solutions approach the Riemann solutions at algebraic rates astime increases. Moreover, a new discovery in this paper is that the differencebetween the two periodic perturbations at two infinities may generate aconstant shift on the background shock wave, which is different from the resultin [11], where the two periodic perturbations are the same.

On Riemann solutions under different initial periodic perturbations at two infinities for 1-d scalar convex conservation laws