Stability of planar rarefaction waves for scalar viscous conservation law under periodic perturbations

The large time behavior of the solutions to a multi-dimensional viscous conservation law is considered in this paper. It is shown that the solution time-asymptotically tends to the planar rarefaction wave if the initial perturbations are multi-dimensional periodic. The time-decay rate is also obtained. Moreover, a Gagliardo-Nirenberg type inequality is established in the domain $\mathbb{R}\times {\mathbb{T}}^{n-1}(n\ge 2)$, where ${\mathbb{T}}^{n-1}$ is the $n-1$-dimensional torus.