First of all we note that if u is continuously differentiable, then D β t u(0) = 0 and consequently (1.1) implies Au 0 = f (0). Therefore if we take as initial **data** U 0 = u 0 , the correction function vanishesĜ ≡ 0. Next, for u to be smooth, in general f is not smooth but behaves as
f (t) ∼ f (0) + c 1 t 1−β
asymptotically as t → 0. To understand the origin of this singularity, simply substitute u = u(0) + u ′ (0)t in the fractional equation (1.1).