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

P. Álvarez-Caudevilla; J. D. Evans; V. A. Galaktionov

doi:10.3934/dcds.2015.35.807

Article Contents

2015, Volume 35, Issue 3: 807-827. Doi: 10.3934/dcds.2015.35.807

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

1.
Universidad Carlos III de Madrid, Av. Universidad 30, 28911-Leganés
2.
Department of Mathematical Sciences, University of Bath, Bath BA2 7AY

Received: January 31, 2013

Revised: June 30, 2014

Published: February 2015

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Abstract

Fundamental global similarity solutions of the standard form $$ u_\gamma(x,t) = t^{-\alpha_\gamma} f_\gamma(y),\,\,\mbox{with the rescaled variable}\,\,\, y= x/{t^{\beta_\gamma}}, \,\, \beta_\gamma= \frac {1-n \alpha_\gamma}{10}, $$ where $\alpha_\gamma>0$ are real nonlinear eigenvalues ($\gamma$ is a multiindex in $\mathbb{R}^N$) of the tenth-order thin film equation (TFE-10) \begin{eqnarray*} \label{i1a} u_{t} = \nabla \cdot (|u|^{n} \nabla \Delta^4 u) \quad in \quad \mathbb{R}^N \times \mathbb{R}_+ \,, \quad n>0, (0.1) \end{eqnarray*} are studied. The present paper continues the study began in [1]. 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 $y \to \infty$ are described. In particular, for $N=1$, the interfaces are shown to diverge as follows: $$ |x_0(t)| \sim 10 ( \frac{1}{n}\sec ( \frac{4\pi}{9} ) )^{\frac 9{10}} t^{\frac 1{10}} \to \infty as n \to 0^+. $$
(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 $\{f_\gamma\}_{|\gamma| \ge 0}$ by a combination of numerical and analytical methods.

Keywords:

Thin film equation,
the Cauchy problem,
source-type global similarity solutions of changing sign.

Mathematics Subject Classification: 35G20, 35K65, 35K35, 37K50.

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References

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