Asymptotic Properties of Entropy Solutions to Fractal Burgers Equation

We study properties of solutions of the initial value problem for the nonlinear and nonlocal equation $u_t+(-\partial^2_x)^{\alpha/2}u+uu_x=0$ with $\alpha\in(0,1]$, supplemented with an initial datum approaching the constant states $u_\pm$ ($u_-<u_+$) as $x\to\pm\infty$, respectively. It was shown by Karch, Miao, and Xu [SIAM J. Math. Anal., 39 (2008), pp. 1536–1549] that, for $\alpha\in(1,2)$, the large time asymptotics of solutions is described by rarefaction waves. The goal of this paper is to show that the asymptotic profile of solutions changes for $\alpha\leq1$. If $\alpha=1$, there exists a self-similar solution to the equation which describes the large time asymptotics of other solutions. In the case $\alpha\in(0,1)$, we show that the nonlinearity of the equation is negligible in the large time asymptotic expansion of solutions.