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 derivative(2)(x))(alpha/2)u + uu(x) = 0 with alpha is an element of (0, 1], supplemented with an initial datum approaching the constant states u(+/-) (u(-) <u(+)) as x -> +/-infinity, respectively. It was shown by Karch, Miao, and Xu [S1AM J. Math. Anal., 39 (2008), pp. 1536-1549] that, for alpha is an element of (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 <= 1. 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 is an element of (0, 1), we show that the nonlinearity of the equation is negligible in the large time asymptotic expansion of solutions.