Is there a relation between abs(fft(u.^2)) and abs((fft(u)).^2
Is there a relation?

Generally no. Squaring is a highly non-linear operation and completely changes the amplitude statistics and spectrum of the original signal. A simple example: if your input consists of a few sine waves of different frequency, the output will be all positive and the spectrum will contain lines are all possible combinations of sum and difference of the original frequencies. The bandwidth will also double which in the discrete domain will often result in aliasing.

has a power law +1

I have no idea what that means.