  We study the long-time asymptotics of the doubly nonlinear diffusion equation
$\rho_t={div}({|\nabla\rho^m|^{p-2}\nabla\rho^m})$ in $\RR^n$, in the range
$\frac{n-p}{n(p-1)}<m>\frac{n-p+1}{n(p-1)}$ and $1p\infty$ where the mass of
the solution is conserved, but the associated energy functional is not
displacement convex. Using a linearisation of the equation, we prove an
$L^1$-algebraic decay of the non-negative solution to a Barenblatt-type
solution, and we estimate its rate of convergence. We then derive the nonlinear
stability of the solution by means of some comparison method between the
nonlinear equation and its linearisation. Our results cover the exponent
interval $\frac{2n}{n+1} p\frac{2n+1}{n+1}$ where a rate of convergence towards
self-similarity was still unknown for the $p$-Laplacian equation.
