    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 . 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
##### References:
  P. Álvarez-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, ().   Google Scholar  P. Álvarez-Caudevilla, J. D. Evans and V. A. Galaktionov, Self-similar Blow-up and its Global Extension for an Unstable Tenth-Order Thin Film Equation,, in preparation., ().   Google Scholar  P. Álvarez-Caudevilla and V. A. Galaktionov, Local bifurcation-branching analysis of global and "blow-up" patterns for a fourth-order thin film equation,, Nonlinear Differ. Equat. Appl., 18 (2011), 483.  doi: 10.1007/s00030-011-0105-6.  Google Scholar  P. Álvarez-Caudevilla and V. A. Galaktionov, Well-posedness of the Cauchy problem for a fourth-order thin film equation via regularization approaches,, submitted., ().   Google Scholar  F. Bernis and A. Friedman, Higher order nonlinear degenerate parabolic equations,, J. Differ. Equat., 83 (1990), 179.  doi: 10.1016/0022-0396(90)90074-Y.  Google Scholar  M. S. Birman and M. Z. Solomjak, Spectral Theory of Self-Adjoint Operators in Hilbert Spaces,, D. Reidel, (1987).  doi: 10.1007/978-94-009-4586-9. Google Scholar  M. Chaves and V. A. Galaktionov, On source-type solutions and the Cauchy problem for a doubly degenerate sixth-order thin film equation. I. Local oscillatory properties,, Nonlinear Anal., 72 (2010), 4030.  doi: 10.1016/j.na.2010.01.034.  Google Scholar  K. Deimling, Nonlinear Functional Analysis,, Springer-Verlag, (1985).  doi: 10.1007/978-3-662-00547-7.  Google Scholar  Yu. V. Egorov, V. A. Galaktionov, V. A. Kondratiev and S. I. Pohozaev, Global solutions of higher-order semilinear parabolic equations in the supercritical range,, Adv. Differ. Equat., 9 (2004), 1009. Google Scholar  J. D. Evans, V. A. Galaktionov and J. R. King, Blow-up similarity solutions of the fourth-order unstable thin film equation,, Euro. J. Appl. Math., 18 (2007), 195.  doi: 10.1017/S0956792507006900.  Google Scholar  J. D. Evans, V. A. Galaktionov and J. R. King, Source-type solutions of the fourth-order unstable thin film equation,, Euro. J. Appl. Math., 18 (2007), 273.  doi: 10.1017/S0956792507006912.  Google Scholar  J. D. Evans, V. A. Galaktionov and J. R. King, Unstable sixth-order thin film equation I. Blow-up similarity solutions,, Nonlinearity, 20 (2007), 1799.  doi: 10.1088/0951-7715/20/8/002.  Google Scholar  J. D. Evans, V. A. Galaktionov and J. R. King, Unstable sixth-order thin film equation II. Global similarity patterns,, Nonlinearity, 20 (2007), 1843.  doi: 10.1088/0951-7715/20/8/003.  Google Scholar  V. A. Galaktionov, Geometric Sturmian Theory of Nonlinear Parabolic Equations and Applications,, Chapman & Hall/CRC, (2004).  doi: 10.1201/9780203998069.  Google Scholar  V. A. Galaktionov, Countable branching of similarity solutions of higher-order porous medium type equations,, Adv. Differ. Equat., 13 (2008), 641. Google Scholar  V. A. Galaktionov, Very singular solutions for thin film equations with absorption,, Studies Appl. Math., 124 (2010), 39.  doi: 10.1111/j.1467-9590.2009.00461.x.  Google Scholar  V. A. Galaktionov, E. Mitidieri and S. I. Pohozaev, Variational approach to complicated similarity solutions of higher-order nonlinear evolution equations of parabolic, hyperbolic, and nonlinear dispersion types,, In Sobolev Spaces in Mathematics. II, 9 (2009), 147.  doi: 10.1007/978-0-387-85650-6_8.  Google Scholar  V. A. Galaktionov, E. Mitidieri and S. I. Pohozaev, Variational approach to complicated similarity solutions of higher-order nonlinear PDEs. II,, Nonl. Anal.: RWA, 12 (2011), 2435.  doi: 10.1016/j.nonrwa.2011.03.001.  Google Scholar  M. A. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations,, Pergamon Press, (1964). Google Scholar  M. A. Krasnosel'skii and P. P. Zabreiko, Geometrical Methods of Nonlinear Analysis,, Springer-Verlag, (1984).  doi: 10.1007/978-3-642-69409-7.  Google Scholar  C. Liu, Qualitative properties for a sixth-order thin film equation,, Mathematical Modelling and Analysis, 15 (2010), 457.  doi: 10.3846/1392-6292.2010.15.457-471.  Google Scholar  X. Liu and C. Qu, Existence and blow-up of weak solutions for a sixth-order equation related to thin solid films,, Nonlinear Anal. Real World Appl., 11 (2010), 4214.  doi: 10.1016/j.nonrwa.2010.05.008.  Google Scholar  L. Perko, Differential Equations and Dynamical Systems,, Springer Verlag, (1991).  doi: 10.1007/978-1-4684-0392-3.  Google Scholar  M. A. Vainberg and V. A. Trenogin, Theory of Branching of Solutions of Non-Linear Equations,, Noordhoff Int. Publ., (1974). Google Scholar

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##### References:
  P. Álvarez-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, ().   Google Scholar  P. Álvarez-Caudevilla, J. D. Evans and V. A. Galaktionov, Self-similar Blow-up and its Global Extension for an Unstable Tenth-Order Thin Film Equation,, in preparation., ().   Google Scholar  P. Álvarez-Caudevilla and V. A. Galaktionov, Local bifurcation-branching analysis of global and "blow-up" patterns for a fourth-order thin film equation,, Nonlinear Differ. Equat. Appl., 18 (2011), 483.  doi: 10.1007/s00030-011-0105-6.  Google Scholar  P. Álvarez-Caudevilla and V. A. Galaktionov, Well-posedness of the Cauchy problem for a fourth-order thin film equation via regularization approaches,, submitted., ().   Google Scholar  F. Bernis and A. Friedman, Higher order nonlinear degenerate parabolic equations,, J. Differ. Equat., 83 (1990), 179.  doi: 10.1016/0022-0396(90)90074-Y.  Google Scholar  M. S. Birman and M. Z. Solomjak, Spectral Theory of Self-Adjoint Operators in Hilbert Spaces,, D. Reidel, (1987).  doi: 10.1007/978-94-009-4586-9. Google Scholar  M. Chaves and V. A. Galaktionov, On source-type solutions and the Cauchy problem for a doubly degenerate sixth-order thin film equation. I. Local oscillatory properties,, Nonlinear Anal., 72 (2010), 4030.  doi: 10.1016/j.na.2010.01.034.  Google Scholar  K. Deimling, Nonlinear Functional Analysis,, Springer-Verlag, (1985).  doi: 10.1007/978-3-662-00547-7.  Google Scholar  Yu. V. Egorov, V. A. Galaktionov, V. A. Kondratiev and S. I. Pohozaev, Global solutions of higher-order semilinear parabolic equations in the supercritical range,, Adv. Differ. Equat., 9 (2004), 1009. Google Scholar  J. D. Evans, V. A. Galaktionov and J. R. King, Blow-up similarity solutions of the fourth-order unstable thin film equation,, Euro. J. Appl. Math., 18 (2007), 195.  doi: 10.1017/S0956792507006900.  Google Scholar  J. D. Evans, V. A. Galaktionov and J. R. King, Source-type solutions of the fourth-order unstable thin film equation,, Euro. J. Appl. Math., 18 (2007), 273.  doi: 10.1017/S0956792507006912.  Google Scholar  J. D. Evans, V. A. Galaktionov and J. R. King, Unstable sixth-order thin film equation I. Blow-up similarity solutions,, Nonlinearity, 20 (2007), 1799.  doi: 10.1088/0951-7715/20/8/002.  Google Scholar  J. D. Evans, V. A. Galaktionov and J. R. King, Unstable sixth-order thin film equation II. Global similarity patterns,, Nonlinearity, 20 (2007), 1843.  doi: 10.1088/0951-7715/20/8/003.  Google Scholar  V. A. Galaktionov, Geometric Sturmian Theory of Nonlinear Parabolic Equations and Applications,, Chapman & Hall/CRC, (2004).  doi: 10.1201/9780203998069.  Google Scholar  V. A. Galaktionov, Countable branching of similarity solutions of higher-order porous medium type equations,, Adv. Differ. Equat., 13 (2008), 641. Google Scholar  V. A. Galaktionov, Very singular solutions for thin film equations with absorption,, Studies Appl. Math., 124 (2010), 39.  doi: 10.1111/j.1467-9590.2009.00461.x.  Google Scholar  V. A. Galaktionov, E. Mitidieri and S. I. Pohozaev, Variational approach to complicated similarity solutions of higher-order nonlinear evolution equations of parabolic, hyperbolic, and nonlinear dispersion types,, In Sobolev Spaces in Mathematics. II, 9 (2009), 147.  doi: 10.1007/978-0-387-85650-6_8.  Google Scholar  V. A. Galaktionov, E. Mitidieri and S. I. Pohozaev, Variational approach to complicated similarity solutions of higher-order nonlinear PDEs. II,, Nonl. Anal.: RWA, 12 (2011), 2435.  doi: 10.1016/j.nonrwa.2011.03.001.  Google Scholar  M. A. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations,, Pergamon Press, (1964). Google Scholar  M. A. Krasnosel'skii and P. P. Zabreiko, Geometrical Methods of Nonlinear Analysis,, Springer-Verlag, (1984).  doi: 10.1007/978-3-642-69409-7.  Google Scholar  C. Liu, Qualitative properties for a sixth-order thin film equation,, Mathematical Modelling and Analysis, 15 (2010), 457.  doi: 10.3846/1392-6292.2010.15.457-471.  Google Scholar  X. Liu and C. Qu, Existence and blow-up of weak solutions for a sixth-order equation related to thin solid films,, Nonlinear Anal. Real World Appl., 11 (2010), 4214.  doi: 10.1016/j.nonrwa.2010.05.008.  Google Scholar  L. Perko, Differential Equations and Dynamical Systems,, Springer Verlag, (1991).  doi: 10.1007/978-1-4684-0392-3.  Google Scholar  M. A. Vainberg and V. A. Trenogin, Theory of Branching of Solutions of Non-Linear Equations,, Noordhoff Int. Publ., (1974). Google Scholar
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