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

On the rate of convergence to equilibrium for the homogeneous Landau equation with soft potentials