The Cauchy problem for a tenth-order thin film equation II. Oscillatory source-type and fundamental similarity solutions

TL;DR

This paper investigates the fundamental similarity solutions of a tenth-order thin film equation, focusing on oscillatory behaviors, eigenvalues, and the effects of the parameter n approaching zero, combining analytical and numerical methods.

Contribution

It extends previous work by analyzing oscillatory solutions, eigenvalues, and interface behaviors of the tenth-order thin film equation, especially as n approaches zero.

Findings

Interfaces diverge as n approaches zero in 1D case.

Oscillatory structures are present near interfaces for fixed n.

Global eigenfunctions are characterized through numerical and analytical methods.

Abstract

Fundamental global similarity solutions of the standard form u_\g(x,t)=t^{-\a_\g} f_\g(y), with the rescaled variable y= x/{t^{\b_\g}}, \b_\g= \frac {1-n \a_\g}{10}, where \a_\g>0 are real nonlinear eigenvalues (\g is a multiindex in R^N) of the tenth-order thin film equation (TFE-10) u_{t} = \nabla \cdot(|u|^{n} \n \D^4 u) in R^N \times R_+, n>0, are studied. The present paper continues the study began by the authors in the previous paper P. Alvarez-Caudevilla, J.D.Evans, and V.A. Galaktionov, The Cauchy problem for a tenth-order thin film equation I. Bifurcation of self-similar oscillatory fundamental solutions, Mediterranean Journal of Mathematics, No. 4, Vol. 10 (2013), 1759-1790. Thus, the following questions are also under scrutiny: (I) Further study of the limit n \to 0, where the behaviour of finite interfaces and solutions as y \to infinity are described. In particular, for N=1, the interfaces are shown to diverge as follows: |x_0(t)| \sim 10 \left( \frac{1}{n}\sec\left( \frac{4\pi}{9} \right) \right)^{\frac 9{10}} t^{\frac 1{10}} \to \infty as n \to 0^+. (II) For a fixed n \in (0, \frac 98), oscillatory structures of solutions near interfaces. (III) Again, for a fixed n \in (0, \frac 98), global structures of some nonlinear eigenfunctions \{f_\g\}_{|\g| \ge 0} by a combination of numerical and analytical methods.