  In this paper, we establish the existence of the unique global-in-time
classical solutions to the multi-component BGK model suggested in
\cite{mixmodel} when the initial data is a small perturbation of global
equilibrium. For this, we carefully analyze the dissipative nature of the
linearized multi-component relaxation operator, and observe that the partial
dissipation from the intra-species and the inter-species linearized relaxation
operators are combined in a complementary manner to give rise to the desired
dissipation estimate of the model We also observe that the convergence rate of
the distribution function increases as the momentum-energy interchange rate
between the different components of the gas increases.
