Critical nonlinearity exponent and self-similar asymptotics for Lévy conservation laws

Abstract

Nonlocal conservation laws of the form

where $-\mathcal{L}$ is the generator of a Lévy semigroup on $L^1(R^n)$, are encountered in continuum mechanics as model equations with anomalous diffusion. They are generalizations of the classical Burgers equation. We study the critical case when the diffusion and nonlinear terms are balanced, e.g. $\mathcal{L}\sim (-\Delta {)}^{\alpha /2}$, $1<\alpha <2$, $f(s)\sim s|s{|}^{r-1}$, $r=1+(\alpha -1)/n$. The results include decay rates of solutions and their genuinely nonlinear asymptotic behavior as time $t$ tends to infinity, determined by self-similar source solutions.