The Cauchy problem for tenth-order thin film equation I. Bifurcation of oscillatory fundamental solutions

TL;DR

This paper investigates the bifurcation of oscillatory fundamental solutions for the tenth-order thin film equation as the parameter n approaches zero, using spectral theory of the associated linear poly-harmonic operator.

Contribution

It introduces a spectral approach to analyze bifurcations of solutions in the tenth-order thin film equation as n tends to zero.

Findings

Identification of bifurcation points for oscillatory solutions

Analysis of the limit n → 0^+ using spectral theory

Characterization of fundamental solutions' behavior near bifurcation

Abstract

Fundamental global similarity solutions of the tenth-order thin film equation u_{t} = \nabla \cdot(|u|^{n} \n \D^4 u) in R^N \times R_+, where n>0 are studied. The main approach consists in passing to the limit n \to 0^+ by using Hermitian non-self-adjoint spectral theory corresponding to the rescaled linear poly-harmonic equation u_t= \D^5 u in R^N \times \re_+.