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

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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

P. Álvarez-Caudevilla ^1, , J. D. Evans ^2, and V. A. Galaktionov ^2,

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

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

Mathematics Subject Classification: 35G20, 35K65, 35K35, 37K5.

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:

[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, (). Google Scholar
[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., (). Google Scholar
[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. doi: 10.1007/s00030-011-0105-6. Google Scholar
[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., (). Google Scholar
[5]	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
[6]	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
[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. doi: 10.1016/j.na.2010.01.034. Google Scholar
[8]	K. Deimling, Nonlinear Functional Analysis,, Springer-Verlag, (1985). doi: 10.1007/978-3-662-00547-7. Google Scholar
[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. Google Scholar
[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. doi: 10.1017/S0956792507006900. Google Scholar
[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. doi: 10.1017/S0956792507006912. Google Scholar
[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. doi: 10.1088/0951-7715/20/8/002. Google Scholar
[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. doi: 10.1088/0951-7715/20/8/003. Google Scholar
[14]	V. A. Galaktionov, Geometric Sturmian Theory of Nonlinear Parabolic Equations and Applications,, Chapman & Hall/CRC, (2004). doi: 10.1201/9780203998069. Google Scholar
[15]	V. A. Galaktionov, Countable branching of similarity solutions of higher-order porous medium type equations,, Adv. Differ. Equat., 13 (2008), 641. Google Scholar
[16]	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*
[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, 9 (2009), 147. doi: 10.1007/978-0-387-85650-6_8. Google Scholar
[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. doi: 10.1016/j.nonrwa.2011.03.001. Google Scholar
[19]	M. A. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations,, Pergamon Press, (1964). Google Scholar
[20]	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
[21]	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
[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. doi: 10.1016/j.nonrwa.2010.05.008. Google Scholar
[23]	L. Perko, Differential Equations and Dynamical Systems,, Springer Verlag, (1991). doi: 10.1007/978-1-4684-0392-3. Google Scholar
[24]	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:

[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, (). Google Scholar
[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., (). Google Scholar
[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. doi: 10.1007/s00030-011-0105-6. Google Scholar
[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., (). Google Scholar
[5]	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
[6]	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
[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. doi: 10.1016/j.na.2010.01.034. Google Scholar
[8]	K. Deimling, Nonlinear Functional Analysis,, Springer-Verlag, (1985). doi: 10.1007/978-3-662-00547-7. Google Scholar
[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. Google Scholar
[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. doi: 10.1017/S0956792507006900. Google Scholar
[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. doi: 10.1017/S0956792507006912. Google Scholar
[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. doi: 10.1088/0951-7715/20/8/002. Google Scholar
[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. doi: 10.1088/0951-7715/20/8/003. Google Scholar
[14]	V. A. Galaktionov, Geometric Sturmian Theory of Nonlinear Parabolic Equations and Applications,, Chapman & Hall/CRC, (2004). doi: 10.1201/9780203998069. Google Scholar
[15]	V. A. Galaktionov, Countable branching of similarity solutions of higher-order porous medium type equations,, Adv. Differ. Equat., 13 (2008), 641. Google Scholar
[16]	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*
[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, 9 (2009), 147. doi: 10.1007/978-0-387-85650-6_8. Google Scholar
[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. doi: 10.1016/j.nonrwa.2011.03.001. Google Scholar
[19]	M. A. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations,, Pergamon Press, (1964). Google Scholar
[20]	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
[21]	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
[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. doi: 10.1016/j.nonrwa.2010.05.008. Google Scholar
[23]	L. Perko, Differential Equations and Dynamical Systems,, Springer Verlag, (1991). doi: 10.1007/978-1-4684-0392-3. Google Scholar
[24]	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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