  The present paper considers a homogeneous bubble inside an unbounded
polytropic compressible liquid with viscosity. The system is governed by the
Navier-Stokes equation with free boundary which is determined by the kinematic
and dynamic boundary conditions on the bubble-liquid interface. The global
existence of solution is proved, and the $\dot{H}^1$ asymptotic stability of
the spherical equilibrium in terms of viscous damping together with a explicit
decay rate is given in bare energy methods.
