  Our focus is on the fast diffusion equation $\partial_t u=\Delta_p u$ with
$p<2$ in the whole Euclidean space of dimension $N\geq 2$. The properties of
the solutions to the $p$-Laplace Cauchy problem change in several special
values of the parameter $p$.
  In the range of $p$ when mass is conserved, non-negative and integrable
solutions behave like the Barenblatt (or fundamental) solutions for large
times. By making use of the entropy method, we establish the polynomial rates
of the convergence in the uniform relative error for a natural class of initial
data. The convergence was established in the literature for $p$ close to $2$,
but no rates were available. In particular, we allow for the values of $p$, for
which the entropy is not displacement convex, as we do not apply the optimal
transportation tools.
  We approach the issue of long-term asymptotics of the gradients of solutions.
In fact, in the case of the radial initial datum, we provide also polynomial
rates of the uniform convergence in the relative error of radial derivatives of
solutions for $\frac{2N}{N+2}<p<2$.
  Finally, providing an analysis of needed properties of solutions for the
entropy method to work, we open the question on the full description of the
basin of attraction of the Bareblatt solutions for $p$ close to $1$.
