  We investigate in this work the rate of convergence to equilibrium of
solutions to the spatially homogeneous Landau equation with soft potentials.
Firstly, we prove a polynomial in time convergence using an entropy method with
some new a priori estimates. Finally we prove an exponential in time
convergence towards the equilibrium with the optimal rate given by the spectral
gap of the associated linearized operator, combining new decay estimates for
the semigroup generated by the linearized Landau operator in weighted
$L^p$-spaces togheter with the polynomial decay described above.
