  We study the dynamic relaxation to equilibrium of the 1D dissipative
Boltzmann equation with Maxwell interactions in classical $H^s$ Sobolev spaces.
In addition, we present a spectral shrinkage analysis and spectral gap
estimates for the linearised 1D dissipative Boltzmann operator with such
interactions. Based on this study, we explore the convergence in $H^s$ and
$L^{1}$ spaces for the linear and nonlinear models. This study extends
classical results found in the literature given for spaces with weak
topologies.
