Electron. J. Diff. Equ., Vol. 2015 (2015), No. 90, pp. 1-29.

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

Pablo Alvarez-Caudevilla, Victor A. Galaktionov

Abstract:
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 fourth-order thin film equation
$$ u_t=-\nabla \cdot (|u|^n \nabla \Delta u) \quad \text{in }\mathbb{R}^N \times \mathbb{R}_ + \text{where }n>0,\; N \ge 1 $$
can be obtained by a continuous deformation (a homotopy path) as $n \to 0^+$. This is done by reducing to similarity solutions (given by eigenfunctions of a rescaled linear operator $\mathbf{B}$) of the classic bi-harmonic equation
$$ u_t = - \Delta^2 u \quad\text{in }\mathbb{R}^N \times \mathbb{R}_ +, \text{ where } \mathbf{B}=-\Delta^2 +\frac 14 y \cdot \nabla+ \frac N4 I. $$
This approach leads to a countable family of various global similarity patterns of the thin film equation, and describes their oscillatory sign-changing behav iour by using the known asymptotic properties of the fundamental solution of bi-harmonic equation. The branching from $n=0^+$ for thin film equation requires Hermitian spectral theory for a pair $\{\mathbf{B}, \mathbf{B}^*\}$ of non-self adjoint operators and leads to a number of difficult mathematical problems. These include, as a key part, the problem of multiplicity of solutions, which is under particular scrutiny.

Submitted April 11, 2014. Published April 10, 2015.
Math Subject Classifications: 35K55, 35B32, 35G20, 35K41, 35K65.
Key Words: Thin film equation; Cauchy problem; source-type similarity solutions; finite interfaces; oscillatory sign-changing behaviour; Hermitian spectral theory; branching.

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Pablo Alvarez-Caudevilla
Universidad Carlos III de Madrid
Av. Universidad 30, 28911-Leganes, Spain
email: pacaudev@math.uc3m.es Phone: +34-916249099
Victor A. Galaktionov
Department of Mathematical Sciences
University of Bath
Bath BA2 7AY, UK
email: vag@maths.bath.ac.uk Phone: +44-1225826988

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