Branching analysis of a countable family of global similarity solutions of a fourth-order thin film equation

The main goal in this article is to justify that source-type and other global-in-time similarity solutions of the Cauchy problem for the fourthorder thin film equation can be obtained by a continuous deformation (a homotopy path). This is done by reducing to similarity solutions (given by eigenfunctions of a rescaled linear operator B) of the classic bi-harmonic equation This approach leads to a countable family of various global similarity patterns of the thin film equation, and describes their oscillatory sign-changing behaviour by using the known asymptotic properties of the fundamental solution of bi-harmonic equation. These include, as a key part, the problem of multiplicity of solutions, which is under particular scrutiny.This work was partially supported by the Ministry of Economy and Competitiveness of Spain under research project MTM2012-33258