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

Alibaud, Nathael; Imbert, Cyril; Karch, Grzegorz

SIAM J. Math. Anal. 42, No. 1, 354-376 (2010).

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.”

Reviewer: Nasser-eddine Tatar (Dhahran)

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

Keywords:

fractal Burgers equation; asymptotic behavior of solutions; self-similar solutions; entropy solutions

Citations:

Zbl 1154.35080