Far field asymptotics of solutions to convection equation with anomalous diffusion

TL;DR

This paper investigates the large-distance behavior of solutions to a convection equation with fractional diffusion, providing asymptotic expansions for solutions with rapidly decaying initial data.

Contribution

It derives the far-field asymptotics of solutions to a convection equation with anomalous diffusion for <, under polynomial growth conditions on the nonlinearity.

Findings

Asymptotic expansion of solutions as |x|  

Characterization of decay rates at infinity

Extension to nonlinear fractional diffusion equations

Abstract

The initial value problem for the conservation law $\partial_t u+(-\Delta)^{\alpha/2}u+\nabla \cdot f(u)=0$ is studied for $\alpha\in (1,2)$ and under natural polynomial growth conditions imposed on the nonlinearity. We find the asymptotic expansion as $|x|\to \infty$ of solutions to this equation corresponding to initial conditions, decaying sufficiently fast at infinity.