The HERA data on the proton structure function F-2(x,Q(2)) at very small x
and Q(2) show a dramatic departure of the logarithmic slope partial derivat
ive F-2/partial derivative log Q(2) from theoretical predictions based on t
he DGLAP evolution. We show that the running BFKL approach provides a quant
itative explanation for the observed x and/or Q(2) dependence of partial de
rivative F-2/partial derivative logQ(2). (C) 1999 American Institute of Phy
sics. [S0021-3640(99)00302-3].