Behavior of solutions for radially symmetric solutions for Burgers equation with a boundary corresponding to the rarefaction wave

Abstract

We investigate the large-time behavior of the radially symmetric solution for Burgers equation on the exterior of a small ball in multi-dimensional space, where the boundary data and the data at the far field are prescribed. In a previous paper [1], we showed that, for the case in which the boundary data is equal to $0$ or negative, the asymptotic stability is the same as that for the viscous conservation law. In the present paper, it is proved that if the boundary data is positive, the asymptotic state is a superposition of the stationary wave and the rarefaction wave, which is a new wave phenomenon. The proof is given using a standard $L^{2}$ energy method and the characteristic curve method.