  In this paper, we consider the Boltzmann equation for a polyatomic gas. We
establish that the mild solution to the Boltzmann equation on the torus is
globally well-posed, provided the initial data that satisfy bounded
velocity-weighted $L^{\infty}$ norm and the smallness condition on the initial
relative entropy. Furthermore, we also study the asymptotic behavior of
solutions, converging to the global Maxwellian with an exponential rate. A key
point in the proof is to develop the pointwise estimate on the gain term of
non-linear collision operator for Gr\"{o}nwall's argument.
