On Convergence of Solutions of Fractal Burgers Equation toward Rarefaction Waves

In the paper, the large time behavior of solutions of the Cauchy problem for the one dimensional fractal Burgers equation ut + (−∂ 2 x) α/2u+ uux = 0 with α ∈ (1, 2) is studied. It is shown that if the nondecreasing initial datum approaches the constant states u± (u− < u+) as x → ±∞, respectively, then the corresponding solution converges toward the rarefaction wave, i.e. the unique entropy solution of the Riemann problem for the nonviscous Burgers equation. AMS Subject Classification 2000: 60J60, 35B40, 35K55, 35Q53.