## Asymptotic properties of entropy solutions to fractal Burgers equation.(English)Zbl 1225.35026

Aurthors’ abstract: “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 G. Karch, C. Miao and X. Xu [SIAM J. Math. Anal. 39, No. 5, 1536–1549 (2008; Zbl 1154.35080)] 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.”

### MSC:

 35B40 Asymptotic behavior of solutions to PDEs 35K15 Initial value problems for second-order parabolic equations 35R11 Fractional partial differential equations 35K58 Semilinear parabolic equations 35C06 Self-similar solutions to PDEs

Zbl 1154.35080
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