  We consider the dynamics of a gas bubble immersed in an incompressible fluid
of fixed temperature, and focus on the relaxation of an expanding and
contracting spherically symmetric bubble due to thermal effects. We study two
models, both systems of PDEs with an evolving free boundary: the full
mathematical model as well as an approximate model, arising for example in the
study of sonoluminescence. For fixed physical parameters (surface tension of
the gas-liquid interface, liquid viscosity, thermal conductivity of the gas,
etc.), both models share a family of spherically symmetric equilibria, smoothly
parametrized by the mass of the gas bubble. Our main result concerns the
approximate model. We prove the nonlinear asymptotic stability of the manifold
of equilibria with respect to small spherically symmetric perturbations. The
rate of convergence is exponential in time. To prove this result we first prove
a weak form of nonlinear asymptotic stability -- with no explicit rate of
time-decay -- using the energy dissipation law, and then, via a center manifold
analysis, bootstrap the weak time-decay to exponential time-decay.
  We also study the uniqueness of the family of spherically symmetric
equilibria within each model. The family of spherically symmetric equilibria
captures all regular spherically symmetric equilibria of the approximate
system. However within the full model, this family is embedded in a larger
family of spherically symmetric solutions. For the approximate system, we prove
that all equilibrium bubbles are spherically symmetric, by an application of
Alexandrov's theorem on closed surfaces of constant mean curvature.
