Large time behaviour of solutions to the $N-$dimensional scalar conservation law under periodic perturbations with nonlinear degenerate viscosity

In this paper, we discuss the asymptotic behaviour of the weak solution to the Cauchy problem for the scalar viscous conservation law, with nonlinear Laplacian viscosity. Firstly, we obtain the existence, uniqueness and regularity of solutions when the initial data $u_0\in C^1({\mathbb{R}}^N)\cap W^{1,\mathrm{\infty }}({\mathbb{R}}^N)$. Secondly, when $u_0$ is periodic, we prove the time-decay rate of the periodic solution and its gradient. At last, we study the long-time behaviour of perturbed solution to the Cauchy problem, in which the initial data is a $N-$d periodic perturbation around a planar rarefaction wave and obtain the time-decay rate of the perturbed solution approaching approximate planar rarefaction wave. The proof is given by technical energy methods and iteration technique.