On the trend to equilibrium for some dissipative systems with slowly increasing a priori bounds

We prove convergence to equilibrium with explicit rates for various kinetic equations with relatively bad control of the distribution tails: in partic ular. Boltzmann-type equations with (smoothed) soft potentials. We compensa te the lack of uniform-in-time estimates by the use of precise logarithmic Sobolev-type inequalities, and the assumption that the initial datum decays rapidly at large velocities Our method not only gives explicit results on the times of convergence, but is also able to cover situations in which com pactness arguments apparently do not apply (even mere convergence to equili brium was an open problem for soft potentials).