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On continuous branches of very singular similarity solutions of the stable thin film equation. II – Free-boundary problems

Published online by Cambridge University Press:  21 February 2011

J. D. EVANS  and
V. A. GALAKTIONOV
J. D. EVANS
Affiliation:
Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK email: masjde@maths.bath.ac.uk, vag@maths.bath.ac.uk
V. A. GALAKTIONOV
Affiliation:
Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK email: masjde@maths.bath.ac.uk, vag@maths.bath.ac.uk
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Abstract

We discuss the fourth-order thin film equation with a stable second-order diffusion term, in the context of a standard free-boundary problem with zero height, zero contact angle and zero-flux conditions imposed at an interface. For the first critical exponent where N ≥ 1 is the space dimension, there are continuous sets (branches) of source-type very singular self-similar solutions of the form For p ≠ p0, the set of very singular self-similar solutions is shown to be finite and consists of a countable family of branches (in the parameter p) of similarity profiles that originate at a sequence of critical exponents {pl, l ≥ 0}. At p = pl, these branches appear via a non-linear bifurcation mechanism from a countable set of second kind similarity solutions of the pure thin film equation Such solutions are detected by a combination of linear and non-linear ‘Hermitian spectral theory’, which allows the application of an analytical n-branching approach. In order to connect with the Cauchy problem in Part I, we identify the cauchy problem solutions as suitable ‘limits’ of the free-boundary problem solutions.

Keywords

Stable thin film equationGlobal similarity solutionsAsymptotic behaviourBranchingBifurcationsHermitian spectral theory
Type
Papers
Information
European Journal of Applied Mathematics , Volume 22 , Issue 3 , June 2011 , pp. 245 - 265
Copyright
Copyright © Cambridge University Press 2011

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