Nonlinear asymptotic stability of non-self-similar rarefaction wave for two-dimensional viscous Burgers equation

We investigate the large time behavior of solutions to the two-dimensional viscous Burgers equation $u_t+u{u}_x+u{u}_y=\mathrm{\Delta }u$, toward a non-self-similar rarefaction wave of inviscid Burgers equation with two initial constant states, seperated by a curve $y=\phi (x)$, and prove that the above 2D non-self-similar rarefaction wave is time-asymptotically stable. Furthermore, we also get the decay rate. Both the rarefaction wave strength and the initial perturbation can be large.