We consider the spatially inhomogeneous Boltzmann equation for inelastic hard-spheres, with constant restitution coefficient α ∈ (0, 1), under the thermalization induced by a host medium with fixed e ∈ (0, 1] and a fixed Maxwellian distribution. When the restitution coefficient α is close to 1 we prove existence and uniqueness of global solutions considering the close-toequilibrium regime. We also study the long-time behaviour of these solutions and prove a convergence to equilibrium with an exponential rate.