  In this letter we discuss cosmological first order phase transitions with de
Sitter bubbles nucleating on (inhomogeneous) matter backgrounds. The de Sitter
bubble can be a toy model for an inflationary phase of universes like our own.
Using the thin wall approximation and the Israel junction method we trace the
classical evolution of the formed bubbles within a compound model. We first
address homogeneous ambient space (FRW model) and already find that bubbles
nucleated in a dust dominated background cannot expand. For an inhomogeneous
dust background (LTB model) we describe cases with at least initially expanding
bubbles. Yet, an ensuing passage of the bubble wall through ambient curvature
inhomogeneities remains unnoticed for observers inside the bubble. Notable
effects also for interior observers are found in the case of a rapid background
phase transition in a FRW model.
