Local Bifurcation-Branching Analysis of Global and "Blow-up" Patterns for a Fourth-Order Thin Film Equation

TL;DR

This paper analyzes the bifurcation and branching of solutions, including global and blow-up patterns, for a fourth-order thin film equation, revealing infinite families of similarity solutions derived from eigenfunctions of associated linear operators.

Contribution

It introduces a bifurcation-branching analysis for the fourth-order thin film equation, identifying infinite solution families originating from linear eigenfunctions.

Findings

Existence of infinite families of similarity solutions.

Identification of global and blow-up solution patterns.

Solutions originate from eigenfunctions of linear operators.

Abstract

The quasilinear fourth-order thin film equation in $R^N$ is shown to admit infinite countable families of global and blow-up similarity solutions, which are originated in the limit from eigenfunctions of the corresponding linear non-self-adjoint operators.