  The stability of $q$-Gaussian distributions as particular solutions of the
linear diffusion equation and its generalized nonlinear form,
$\pderiv{P(x,t)}{t} = D \pderiv{^2 [P(x,t)]^{2-q}}{x^2}$, the
\emph{porous-medium equation}, is investigated through both numerical and
analytical approaches. It is shown that an \emph{initial} $q$-Gaussian,
characterized by an index $q_i$, approaches the \emph{final}, asymptotic
solution, characterized by an index $q$, in such a way that the relaxation rule
for the kurtosis evolves in time according to a $q$-exponential, with a
\emph{relaxation} index $q_{\rm rel} \equiv q_{\rm rel}(q)$. In some cases,
particularly when one attempts to transform an infinite-variance distribution
($q_i \ge 5/3$) into a finite-variance one ($q<5/3$), the relaxation towards
the asymptotic solution may occur very slowly in time. This fact might shed
some light on the slow relaxation, for some long-range-interacting many-body
Hamiltonian systems, from long-standing quasi-stationary states to the ultimate
thermal equilibrium state.
