Existence of solutions describing accumulation in a thin-film flow

We consider a third order non-autonomous ODE that arises as a model of fluid accumulation in a two dimensional thin-film flow driven by surface tension and gravity. With the appropriate matching conditions, the equation describes the inner structure of solutions around a stagnation point. In this paper we prove the existence of solutions that satisfy this problem. In order to prove the result we first transform the equation into a four dimensional dynamical system. In this setting the problem consists of finding heteroclinic connections that are the intersection of a two dimensional centre-stable manifold and a three-dimensional centre-unstable one. We then use a shooting argument that takes advantage of the information of the flow in the far-field, part of the analysis also requires the understanding of oscillatory solutions with large amplitude. The far-field is represented by invariant three-dimensional subspaces and the flow on them needs to be understood, most of the necessary results in this regard are obtained in \cite{CV}. This analysis focuses on the understanding of oscillatory solutions and some results are used in the current proof, although the structure of oscillations is somewhat more complicated.