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On continuous branches of very singular similarity solutions of a stable thin film equation. I – The Cauchy problem

Published online by Cambridge University Press:  21 February 2011

J. D. EVANS  and
V. A. GALAKTIONOV
J. D. EVANS
Affiliation:
Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK email: [email protected], [email protected]
V. A. GALAKTIONOV
Affiliation:
Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK email: [email protected], [email protected]
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Abstract

We consider the fourth-order thin film equation, with a stable second-order diffusion term. For the first critical exponent, where N ≥ 1 is the space dimension, the Cauchy problem is shown to admit countable continuous branches of source-type self-similar very singular solutions of the form These solutions are inherently oscillatory in nature and will be shown in Part II to be the limit of appropriate free-boundary problem solutions. For p ≠ p0, the set of very singular solutions is shown to be finite and to be consisting of a countable family of branches (in the parameter p) of similarity profiles that originate at a sequence of critical exponents {pl, l ≥ 0}. At p = pl, these branches appear via a non-linear bifurcation mechanism from a countable set of similarity solutions of the second kind of the pure thin film equation Such solutions are detected by the ‘Hermitian spectral theory’, which allows an analytical n-branching approach. As such, a continuous path as n → 0+ can be constructed from the eigenfunctions of the linear rescaled operator for n = 0, i.e. for the bi-harmonic equation ut = −Δ2u. Numerics are used, wherever appropriate, to support the analysis.

Keywords

Stable thin film equationCauchy problemGlobal similarity solutionsAsymptotic behaviourBranchingBifurcationsHermitian spectral theory
Type
Papers
Information
European Journal of Applied Mathematics , Volume 22 , Issue 3 , June 2011 , pp. 217 - 243
Copyright
Copyright © Cambridge University Press 2011

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References

[1]Bernis, F. (1988) Source-type solutions of fourth order degenerate parabolic equations. In: Ni, W.-M., Peletier, L. A. & Serrin, J. (editors), Proc. Microprogram Nonlinear Diffusion Eqs Equilibrium States, Vol. 1, MSRI Publications, New York, NY, USA, pp. 123–146.CrossRefGoogle Scholar
[2]Bernis, F. & Friedman, A. (1990) Higher order nonlinear degenerate parabolic equations. J. Differ. Equ. 83, 179–206.CrossRefGoogle Scholar
[3]Bernis, F., Hulshof, J. & King, J. R. (2000) Dipoles and similarity solutions of the thin film equation in the half-line. Nonlinearity 13, 413–439.CrossRefGoogle Scholar
[4]Bernis, F. & McLeod, J. B. (1991) Similarity solutions of a higher order nonlinear diffusion equation. Nonlinear Anal. 17, 1039–1068.CrossRefGoogle Scholar
[5]Bernis, F., Peletier, L. A. & Williams, S. M. (1992) Source-type solutions of a fourth order nonlinear degenerate parabolic equation. Nonlinear Anal. Theory Methods Appl. 18, 217–234.CrossRefGoogle Scholar
[6]Bowen, M., Hulshof, J. & King, J. R. (2001) Anomalous exponents and dipole solutions for the thin film equation. SIAM J. Appl. Math. 62, 149–179.Google Scholar
[7]Bowen, M. & Witelski, T. P. (2006) The linear limit of the dipole problem for the thin film equation. SIAM J. Appl. Math. 66, 1727–1748.CrossRefGoogle Scholar
[8]Egorov, Yu. V., Galaktionov, V. A., Kondratiev, V. A. & Pohozaev, S. I. (2004) Asymptotic behaviour of global solutions to higher-order semilinear parabolic equations in the supercritical range. Adv. Difference Equ. 9, 1009–1038.Google Scholar
[9]Evans, J. D., Galaktionov, V. A. & King, J. R. (2007a) Blow-up similarity solutions of the fourth-order unstable thin film equation. Eur. J. Appl. Math. 18, 195–231.CrossRefGoogle Scholar
[10]Evans, J. D., Galaktionov, V. A. & King, J. R. (2007b) Source-type solutions for the fourth-order unstable thin film equation. Eur. J. Appl. Math. 18, 273–321.CrossRefGoogle Scholar
[11]Evans, J. D., Galaktionov, V. A. & King, J. R. (2007c) Unstable sixth-order thin film equation. I. Blow-up similarity solutions; II. Global similarity patterns. Nonlinearity 20, 1799–1841, 1843–1881.CrossRefGoogle Scholar
[12]Evans, J. D., Galaktionov, V. A. & Williams, J. F. (2006) Blow-up and global asymptotics of the limit unstable Cahn–Hilliard equation. SIAM J. Math. Anal. 38, 64–102.CrossRefGoogle Scholar
[13]Ferreira, R. & Bernis, F. (1997) Source-type solutions to thin-film equations in higher dimensions. Eur. J. Appl. Math. 8, 507–534.CrossRefGoogle Scholar
[14]Galaktionov, V. A. (2004) Evolution completeness of separable solutions of non-linear diffusion equations in bounded domains. Math. Methods Appl. Sci. 27, 1755–1770.CrossRefGoogle Scholar
[15]Galaktionov, V. A. (2007) Sturmian nodal set analysis for higher-order parabolic equations and applications. Adv. Difference Equ. 12, 669–720.Google Scholar
[16]Galaktionov, V. A. (2010) Very singular solutions for thin film equations with absorption. Studies Appl. Math. 124, 39–63 (arXiv:0109.3982).CrossRefGoogle Scholar
[17]Galaktionov, V. A. & Harwin, P. J. (2005a) On evolution completeness of nonlinear eigenfunctions for the porous medium equation in the whole space. Adv. Difference Equ. 10, 635–674.Google Scholar
[18]Galaktinov, V. A. & Harwin, P. J. (2005b) Non-uniqueness and global similarity solutions for a higher-order semilinear parabolic equation. Nonlinearity 18, 717–746.CrossRefGoogle Scholar
[19]Galaktionov, V. A. & Harwin, P. J. (2009) On centre subspace behaviour in thin film equations. SIAM J. Appl. Math. 69, 1334–1358 (an earlier preprint in arXiv:0901.3995v1).CrossRefGoogle Scholar
[20]Galaktionov, V. A. & Svirshchevskii, S. R. (2007) Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics, Chapman & Hall/CRC, Boca Raton, FL, USA.Google Scholar
[21]Galaktionov, V. A. & Williams, J. F. (2004) On very singular similarity solutions of a higher-order semilinear parabolic equation. Nonlinearity 17, 1075–1099.CrossRefGoogle Scholar
[22]Gohberg, I., Goldberg, S. & Kaashoek, M. A. (1990) Classes of Linear Operators, Vol. 1; Operator Theory: Advances and Applications, Vol. 49, Birkhäuser Verlag, Basel, Switzerland/Berlin, Germany.Google Scholar
[23]Hulshof, J. (1991) Similarity solutions of the porous medium equation with sign changes. J. Math. Anal. Appl. 157, 75–111.CrossRefGoogle Scholar
[24]Kolmogorov, A. N. & Fomin, S. V. (1976) Elements of the Theory of Functions and Functional Analysis, Nauka, Moscow, Russia.Google Scholar
[25]Krasnosel'skii, M. A. & Zabreiko, P. P. (1984) Geometrical Methods of Nonlinear Analysis, Springer-Verlag, Berlin, Germany.CrossRefGoogle Scholar
[26]Perko, L. (1991) Differential Equations and Dynamical Systems, Springer-Verlag, New York, USA.CrossRefGoogle Scholar
[27]Vainberg, M. A. & Trenogin, V. A. (1974) Theory of Branching of Solutions of Non-Linear Equations, Noordhoff International Publishing, Leiden, Netherlands.Google Scholar
[28]Wu, Z., Zhao, J., Yin, J. & Li, H. (2001) Nonlinear Diffusion Equations, World Scientific Publishing Company, River Edge, NJ, USA.CrossRefGoogle Scholar
[29]Zel'dovich, Ya. B. (1956) The motion of a gas under the action of a short-term pressure shock. Akust. Zh., 2, 28–38; Sov. Phys. Acoust. 2, 25–35.Google Scholar