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In the paper the question - Is the q-Fourier transform of a q-Gaussian aq'-Gaussian (with some q') up to a constant factor? - is studied for the wholerange of $q\in (-\infty, 3)$. This question is connected with applicability ofthe q-Fourier transform in the study of limit processes in nonextensivestatistical mechanics. We prove that the answer is affirmative if and only if q> 1, excluding two particular cases of q<1, namely, q = 1/2 and q = 2/3, whichare also out of the theory valid for q \ge 1. We also discuss some applicationsof the q-Fourier transform to nonlinear partial differential equations such asthe porous medium equation.

Functional-differential equations for $F_q$%-transforms of $q$-Gaussians