# 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 and Continuous Dynamical Systems, 2015, 35 (3) : 807-827. doi: 10.3934/dcds.2015.35.807
##### References:
 [1] 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, accepted. [2] 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. [3] 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), 483-537. doi: 10.1007/s00030-011-0105-6. [4] P. Álvarez-Caudevilla and V. A. Galaktionov, Well-posedness of the Cauchy problem for a fourth-order thin film equation via regularization approaches, submitted. [5] F. Bernis and A. Friedman, Higher order nonlinear degenerate parabolic equations, J. Differ. Equat., 83 (1990), 179-206. doi: 10.1016/0022-0396(90)90074-Y. [6] M. S. Birman and M. Z. Solomjak, Spectral Theory of Self-Adjoint Operators in Hilbert Spaces, D. Reidel, Dordecht/Tokyo, 1987. doi: 10.1007/978-94-009-4586-9. [7] 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), 4030-4048. doi: 10.1016/j.na.2010.01.034. [8] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin/Tokyo, 1985. doi: 10.1007/978-3-662-00547-7. [9] 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), 1009-1038. [10] 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), 195-231. doi: 10.1017/S0956792507006900. [11] 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), 273-321. doi: 10.1017/S0956792507006912. [12] J. D. Evans, V. A. Galaktionov and J. R. King, Unstable sixth-order thin film equation I. Blow-up similarity solutions, Nonlinearity, 20 (2007), 1799-1841. doi: 10.1088/0951-7715/20/8/002. [13] J. D. Evans, V. A. Galaktionov and J. R. King, Unstable sixth-order thin film equation II. Global similarity patterns, Nonlinearity, 20 (2007), 1843-1881. doi: 10.1088/0951-7715/20/8/003. [14] V. A. Galaktionov, Geometric Sturmian Theory of Nonlinear Parabolic Equations and Applications, Chapman & Hall/CRC, Boca Raton, Florida, 2004. doi: 10.1201/9780203998069. [15] V. A. Galaktionov, Countable branching of similarity solutions of higher-order porous medium type equations, Adv. Differ. Equat., 13 (2008), 641-680. [16] V. A. Galaktionov, Very singular solutions for thin film equations with absorption, Studies Appl. Math., 124 (2010), 39-63 (arXiv:0109.3982). doi: 10.1111/j.1467-9590.2009.00461.x. [17] 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, Appl. Anal. and Part. Differ. Equat., Series: Int. Math. Ser., V. Maz'ya Ed., Springer, New York, 9(2009), 147-197. (an earlier preprint: arXiv:0902.1425). doi: 10.1007/978-0-387-85650-6_8. [18] 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), 2435-2466 (arXiv:1103.2643). doi: 10.1016/j.nonrwa.2011.03.001. [19] M. A. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations, Pergamon Press, Oxford/Paris, 1964. [20] M. A. Krasnosel'skii and P. P. Zabreiko, Geometrical Methods of Nonlinear Analysis, Springer-Verlag, Berlin/Tokio, 1984. doi: 10.1007/978-3-642-69409-7. [21] C. Liu, Qualitative properties for a sixth-order thin film equation, Mathematical Modelling and Analysis, 15 (2010), 457-471. doi: 10.3846/1392-6292.2010.15.457-471. [22] 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), 4214-4222. doi: 10.1016/j.nonrwa.2010.05.008. [23] L. Perko, Differential Equations and Dynamical Systems, Springer Verlag, New York, 1991. doi: 10.1007/978-1-4684-0392-3. [24] M. A. Vainberg and V. A. Trenogin, Theory of Branching of Solutions of Non-Linear Equations, Noordhoff Int. Publ., Leiden, 1974.

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##### References:
 [1] 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, accepted. [2] 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. [3] 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), 483-537. doi: 10.1007/s00030-011-0105-6. [4] P. Álvarez-Caudevilla and V. A. Galaktionov, Well-posedness of the Cauchy problem for a fourth-order thin film equation via regularization approaches, submitted. [5] F. Bernis and A. Friedman, Higher order nonlinear degenerate parabolic equations, J. Differ. Equat., 83 (1990), 179-206. doi: 10.1016/0022-0396(90)90074-Y. [6] M. S. Birman and M. Z. Solomjak, Spectral Theory of Self-Adjoint Operators in Hilbert Spaces, D. Reidel, Dordecht/Tokyo, 1987. doi: 10.1007/978-94-009-4586-9. [7] 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), 4030-4048. doi: 10.1016/j.na.2010.01.034. [8] K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin/Tokyo, 1985. doi: 10.1007/978-3-662-00547-7. [9] 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), 1009-1038. [10] 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), 195-231. doi: 10.1017/S0956792507006900. [11] 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), 273-321. doi: 10.1017/S0956792507006912. [12] J. D. Evans, V. A. Galaktionov and J. R. King, Unstable sixth-order thin film equation I. Blow-up similarity solutions, Nonlinearity, 20 (2007), 1799-1841. doi: 10.1088/0951-7715/20/8/002. [13] J. D. Evans, V. A. Galaktionov and J. R. King, Unstable sixth-order thin film equation II. Global similarity patterns, Nonlinearity, 20 (2007), 1843-1881. doi: 10.1088/0951-7715/20/8/003. [14] V. A. Galaktionov, Geometric Sturmian Theory of Nonlinear Parabolic Equations and Applications, Chapman & Hall/CRC, Boca Raton, Florida, 2004. doi: 10.1201/9780203998069. [15] V. A. Galaktionov, Countable branching of similarity solutions of higher-order porous medium type equations, Adv. Differ. Equat., 13 (2008), 641-680. [16] V. A. Galaktionov, Very singular solutions for thin film equations with absorption, Studies Appl. Math., 124 (2010), 39-63 (arXiv:0109.3982). doi: 10.1111/j.1467-9590.2009.00461.x. [17] 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, Appl. Anal. and Part. Differ. Equat., Series: Int. Math. Ser., V. Maz'ya Ed., Springer, New York, 9(2009), 147-197. (an earlier preprint: arXiv:0902.1425). doi: 10.1007/978-0-387-85650-6_8. [18] 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), 2435-2466 (arXiv:1103.2643). doi: 10.1016/j.nonrwa.2011.03.001. [19] M. A. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations, Pergamon Press, Oxford/Paris, 1964. [20] M. A. Krasnosel'skii and P. P. Zabreiko, Geometrical Methods of Nonlinear Analysis, Springer-Verlag, Berlin/Tokio, 1984. doi: 10.1007/978-3-642-69409-7. [21] C. Liu, Qualitative properties for a sixth-order thin film equation, Mathematical Modelling and Analysis, 15 (2010), 457-471. doi: 10.3846/1392-6292.2010.15.457-471. [22] 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), 4214-4222. doi: 10.1016/j.nonrwa.2010.05.008. [23] L. Perko, Differential Equations and Dynamical Systems, Springer Verlag, New York, 1991. doi: 10.1007/978-1-4684-0392-3. [24] M. A. Vainberg and V. A. Trenogin, Theory of Branching of Solutions of Non-Linear Equations, Noordhoff Int. Publ., Leiden, 1974.
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