On Oscillations of Solutions of the Fourth-Order Thin Film Equation Near Heteroclinic Bifurcation Point

TL;DR

This paper investigates the behavior of solutions to the fourth-order thin film equation near a heteroclinic bifurcation point, revealing a transition from oscillatory to non-oscillatory solutions at a critical parameter value.

Contribution

It identifies the precise bifurcation point where solutions change from oscillatory to non-oscillatory, linking bifurcation analysis with solution behavior in thin film equations.

Findings

Solutions become non-oscillatory at the bifurcation point n=1.75987.

Solutions may align with nonnegative solutions of a free-boundary problem.

Transition occurs precisely at the heteroclinic bifurcation point.

Abstract

It is shown that solutions of the Cauchy for the fourth-order thin film equation becomes non-oscillatory and non-sign-changing precisely at and above the heteroclinic bifurcation point n=1.75987..., so that, possibly, these solutions begin to coincide with nonnegative solutions of a standard and well-known free-boundary problem.