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Polynomial loss of memory for maps of the interval with a neutral fixed point
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 |
(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.
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. |
[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. |
[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. |
[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. |
[8] |
K. Deimling, Nonlinear Functional Analysis,, Springer-Verlag, (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.
|
[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. |
[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. |
[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. |
[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. |
[14] |
V. A. Galaktionov, Geometric Sturmian Theory of Nonlinear Parabolic Equations and Applications,, Chapman & Hall/CRC, (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.
|
[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. |
[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. |
[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. |
[19] |
M. A. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations,, Pergamon Press, (1964).
|
[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. |
[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. |
[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. |
[23] |
L. Perko, Differential Equations and Dynamical Systems,, Springer Verlag, (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., (1974).
|
show all references
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. |
[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. |
[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. |
[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. |
[8] |
K. Deimling, Nonlinear Functional Analysis,, Springer-Verlag, (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.
|
[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. |
[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. |
[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. |
[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. |
[14] |
V. A. Galaktionov, Geometric Sturmian Theory of Nonlinear Parabolic Equations and Applications,, Chapman & Hall/CRC, (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.
|
[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. |
[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. |
[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. |
[19] |
M. A. Krasnosel'skii, Topological Methods in the Theory of Nonlinear Integral Equations,, Pergamon Press, (1964).
|
[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. |
[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. |
[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. |
[23] |
L. Perko, Differential Equations and Dynamical Systems,, Springer Verlag, (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., (1974).
|
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