American Institute of Mathematical Sciences

March  2015, 35(3): 807-827. doi: 10.3934/dcds.2015.35.807

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, Spain 2 Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, United Kingdom, United Kingdom

Received  February 2013 Revised  July 2014 Published  October 2014

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.
Citation: P. Álvarez-Caudevilla, J. D. Evans, V. A. Galaktionov. The Cauchy problem for a tenth-order thin film equation II. Oscillatory source-type and fundamental similarity solutions. Discrete & Continuous Dynamical Systems - A, 2015, 35 (3) : 807-827. doi: 10.3934/dcds.2015.35.807
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