Asymptotics toward the planar rarefaction wave for viscous conservation law in two space dimensions

This paper is concerned with the asymptotic behavior of the solution toward the planar rarefaction wave $r(\frac{x}{t})$ connecting $u_{+}$ and $u_{-}$ for the scalar viscous conservation law in two space dimensions. We assume that the initial data $u_{0}(x,y)$ tends to constant states $u_{\pm }$ as $x \rightarrow \pm \infty$, respectively. Then, the convergence rate to $r(\frac{x}{t})$ of the solution $u(t,x,y)$ is investigated without the smallness conditions of $|u_{+}-u_{-}|$ and the initial disturbance. The proof is given by elementary $L^{2}$-energy method.