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 ut+(−∂ x) u+uux = 0 with α ∈ (0, 1], supplemented with an initial datum approaching the constant states u± (u− < u+) as x → ±∞, respectively. It was shown by Karch, Miao & Xu (SIAM J. Math. Anal. 39 (2008), 1536–1549) that, for α ∈ (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 α ≤ 1. If α = 1, there exists a self-similar solution to the equation which describes the large time asymptotics of other solutions. In the case α ∈ (0, 1), we show that the nonlinearity of the equation is negligible in the large time asymptotic expansion of solutions.