  We consider scalar conservation laws with nonlocal diffusion of Riesz-Feller
type such as the fractal Burgers equation. The existence of traveling wave
solutions with monotone decreasing profile has been established recently (in
special cases). We show the local asymptotic stability of these traveling wave
solutions in a Sobolev space setting by constructing a Lyapunov functional.
Most importantly, we derive the algebraic-in-time decay of the norm of such
perturbations with explicit algebraic-in-time decay rates.
