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

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