Far field asymptotics of solutions to convection equation with anomalous diffusion

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.

16 pages