At low temperatures, uniaxial perovskites exhibit a spontaneous polarization with a permanent dipole moment. The broken symmetry state involves an order parameter and dielectric properties of the materials which are strongly dominated by a one-dimensional physics. For the last sixty years there has been growing interest in these crystals for potential applications in communication technology. Very recently, renewed interest to ferroelectric memory devices has been remarkable with particular emphasis on miniaturized devices. This new perspective comes along with great challenges, one of which is the critical size for stable domains in thin ferroelectric crystals. Since the polarization switching involves a pre-existing spontaneous polarization, it is of fundamental importance to address the question of conditions under which polarized domains can develop in a ferroelectric thin film. From previous articles, it has been observed that most studies focus on numerical simulations which is a good approach but for the fact that numerical simulations involve approximation of physical quantities, which is limitation to comparing experimental with theoretical studies. This motivated the authors to look purely at an analytical approach taking advantage of the new mathematical approach in the study of nonlinear systems. In this work, the authors have considered the question from an analytical point of view, focusing on an interesting model introduced by Lu and Cao. We propose an analytical counterpart of the numerical simulations done in this previous study.