  In this paper we show that there exist two different critical exponents for
global small data solutions to the semilinear fractional diffusive equation
with Caputo fractional derivative in time. The second critical exponent appears
if the second data is assumed to be zero. This peculiarity is related to the
fact that the order of the equation is fractional. To prove our result, we
first derive Lr-Lq linear estimates for the solution to the inhomogeneous
linear Cauchy problem and then we apply a contraction argument.
