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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

Citations:

Zbl 1154.35080
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