We study properties of solutions of the initial value problem for the nonlinear and nonlocal equation uₜ+(-∂²ₓ)α/2u+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, and Xu [SIAM J. Math. Anal., 39 (2008), pp. 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.
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Alibaud et al. (2010) studied this question.