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We study the radial relaxation dynamics toward equilibrium and time-periodicpulsating spherically symmetric gas bubbles in an incompressible liquid due tothermal effects. The asymptotic model ([A. Prosperetti, J. Fluid Mech., 1991]and [Z. Biro and J. J. L. Velazquez, SIAM J. Math. Anal., 2000]) is one wherethe pressure within the gas bubble is spatially uniform and satisfies an idealgas law, relating the pressure, density and temperature of the gas. Thetemperature of the surrounding liquid is taken to be constant and the behaviorof the liquid pressure at infinity is prescribed to be constant or periodic intime. In arXiv:2207.04079, for the case where the liquid pressure at infinityis a positive constant, we proved the existence of a one-parameter manifold ofspherical equilibria, parameterized by the bubble mass, and further proved thatit is a nonlinearly and exponentially asymptotically stable center manifold.  In the present article, we first refine the exponential time-decay estimates,via a study of the linearized dynamics subject to the constraint of fixed mass.We obtain, in particular, estimates for the exponential decay rate constant,which highlight the interplay between the effects of thermal diffusivity andthe liquid viscosity.  We then study the nonlinear radial dynamics of the bubble-fluid systemsubject to a pressure field at infinity which is a small-amplitude andtime-periodic perturbation about a positive constant. We prove that nonlinearlyand exponentially asymptotically stable time-periodically pulsating solutionsof the nonlinear (asymptotic) model exist for all sufficiently small forcingamplitudes. The existence of such states is formulated as a fixed point problemfor the Poincar\'e return map, and the existence of a fixed point makes use ofour (constant mass constrained) exponential time-decay estimates of thelinearized problem.

Thermal relaxation toward equilibrium and periodically pulsating spherical bubbles in an incompressible liquid