  In this paper we consider the Boltzmann equation modelling the motion of a
polyatomic gas where the integration collision operator in comparison with the
classical one involves an additional internal energy variable
$I\in\mathbb{R}_+$ and a parameter $\delta\geq 2$ standing for the degree of
freedom. In perturbation framework, we establish the global well-posedness for
bounded mild solutions near global equilibria on torus. The proof is based on
the $L^2\cap L^\infty$ approach. Precisely, we first study the $L^2$ decay
property for the linearized equation, then use the iteration technique for the
linear integral operator to get the linear weighted $L^\infty$ decay, and in
the end obtain $L^\infty$ bounds as well as exponential time decay of solutions
for the nonlinear problem with the help of the Duhamel's principle. Throughout
the proof, we present a careful analysis for treating the extra effect of
internal energy variable $I$ and the parameter $\delta$.
