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August  2018, 38(8): 3913-3938. doi: 10.3934/dcds.2018170

Gradient blow-up for a fourth-order quasilinear Boussinesq-type equation

Pablo Álvarez-Caudevilla 1, , Jonathan D. Evans 2, and  Victor A. Galaktionov 2,

1. 

Universidad Carlos Ⅲ de Madrid, Av. Universidad 30, 28911-Leganés, Spain & Instituto de Ciencias Matemáticas, ICMAT, C/Nicolás Cabrera 15, 28049 Madrid, Spain
2. 

Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK

Received  September 2017 Revised  February 2018 Published  May 2018

Fund Project: The first author was partially supported by the Ministry of Economy and Competitiveness of Spain under research projects RYC-2014-15284 and MTM2016-80618-P.

The Cauchy problem for a fourth-order Boussinesq-type quasilinear wave equation (QWE-4) of the form
$u_{tt} = -(|u|^n u)_{xxxx}\;\;\; \mbox{in}\;\;\; \mathbb{R} × \mathbb{R}_+, \;\;\;\mbox{with a fixed exponent} \, \, \, n>0, $
and bounded smooth initial data, is considered. Self-similar single-point gradient blow-up solutions are studied. It is shown that such singular solutions exist and satisfy the case of the so-called self-similarity of the second type.
Together with an essential and, often, key use of numerical methods to describe possible types of gradient blow-up, a "homotopy" approach is applied that traces out the behaviour of such singularity patterns as
$n \to 0^+$
, when the classic linear beam equation occurs
$u_{tt} = -u_{xxxx}, $
with simple, better-known and understandable evolution properties.
Keywords: Fourth-order quasilinear wave equation, gradient blow-up, selfsimilarity of the second kind.
Mathematics Subject Classification: 41A60, 35C20, 35G20, 35C06.
Citation: Pablo Álvarez-Caudevilla, Jonathan D. Evans, Victor A. Galaktionov. Gradient blow-up for a fourth-order quasilinear Boussinesq-type equation. Discrete & Continuous Dynamical Systems, 2018, 38 (8) : 3913-3938. doi: 10.3934/dcds.2018170
References:
[1]

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.  Google Scholar

[2]

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, 10 (2013), 1761-1792.  doi: 10.1007/s00009-013-0263-3.  Google Scholar

[3]

P. Álvarez-Caudevilla and V. A. Galaktionov, Well-posedness of the Cauchy problem for a fourth-order thin film equation via regularization approaches, Nonlinear Analysis, 121 (2015), 19-35.  doi: 10.1016/j.na.2014.08.002.  Google Scholar

[4]

K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin/Tokyo, 1985. doi: 10.1007/978-3-662-00547-7.  Google Scholar

[5]

J. Eggers and M. Fontelos, The role of self-similarity in singularities of partial differential equations, Nonlinearity, 22 (2009), R1-R44.  doi: 10.1088/0951-7715/22/1/R01.  Google Scholar

[6]

Yu. V. Egorov, V. A. Galaktionov, V. A. Kondratiev and S. I. Pohozaev, Asymptotic behaviour of global solutions to higher-order semilinear parabolic equations in the supercritical range, Adv. Differ. Equat., 9 (2004), 1009-1038.   Google Scholar

[7]

J. D. Evans, V. A. Galaktionov and J. R. King, Unstable sixth-order thin film equation. Ⅰ. Blow-up similarity solutions; Ⅱ. Global similarity patterns, Nonlinearity, 20 (2007), 1799-1841, 1843-1881.  doi: 10.1088/0951-7715/20/8/003.  Google Scholar

[8]

A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, Classification of general Wentzell boundary conditions for fourth order operators in one space dimension, J. Math. Anal. Appl., 333 (2007), 219-235.  doi: 10.1016/j.jmaa.2006.11.058.  Google Scholar

[9]

V. A. Galaktionov, On higher-order viscosity approximations of odd-order nonlinear PDEs, J. Engr. Math., 60 (2008), 173-208.  doi: 10.1007/s10665-007-9146-6.  Google Scholar

[10]

V. A. Galaktionov, Hermitian spectral theory and blow-up patterns for a fourth-order semilinear Boussinesq equation, Stud. Appl. Math., 121 (2008), 395-431.  doi: 10.1111/j.1467-9590.2008.00421.x.  Google Scholar

[11]

V. A. Galaktionov, Formation of shocks in higher-order nonlinear dispersion PDEs: nonuniqueness and nonexistence of entropy, (2009), arXiv: 0902.1635. Google Scholar

[12]

V. A. Galaktionov, Regional, single point, and global blow-up for the fourth-order porous medium type equation with source, J. Partial Differ. Equ., 23 (2010), 105–146, arXiv: 0901.4279.  Google Scholar

[13]

V. A. Galaktionov, Shock waves and compactons for fifth-order nonlinear dispersion equations, European J. Appl. Math., 21 (2010), 1–50. (arXiv: 0902.1632). doi: 10.1017/S0956792509990118.  Google Scholar

[14]

V. A. Galaktionov, Single point gradient blow-up and nonuniqueness for a third-order nonlinear dispersion equation, Stud. Appl. Math., 126 (2011), 103-143.  doi: 10.1111/j.1467-9590.2010.00499.x.  Google Scholar

[15]

V. A. Galaktionov and S. I. Pohozaev, Third-order nonlinear dispersive equations: Shocks, rarefaction, and blow-up waves, Comput. Math. Math. Phys., 48 (2008), 1784-1810.  doi: 10.1134/S0965542508100060.  Google Scholar

[16]

V. A. Galaktionov and S. R. Svirshchevskii, Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics, Chapman & Hall/CRC, Boca Raton, Florida, 2007.  Google Scholar

[17]

M. Inc, New compacton and solitary pattern solutions of the nonlinear modified dispersive Klein-Gordon equations, Chaos, Solitons and Fractals, 33 (2007), 1275-1284.  doi: 10.1016/j.chaos.2006.01.083.  Google Scholar

[18]

M. A. Krasnosel'skii and P. P. Zabreiko, Geometrical Methods of Nonlinear Analysis, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69409-7.  Google Scholar

[19]

A. S. Markus, Introduction to the Spectral Theory of Polynomial Operator Pencils, Transl. Math. Mon., Vol. 71, Amer. Math. Soc., Providence, RI, 1988.  Google Scholar

[20]

M. A. Naimark, Linear Differential Operators, Part Ⅰ, Ungar Publ. Comp., New York, 1968.  Google Scholar

[21]

M. A. Vainberg and V. A. Trenogin, Theory of Branching of Solutions of Non-Linear Equations, Noordhoff Int. Publ., Leiden, 1974.  Google Scholar

[22]

Z. Yan, Constructing exact solutions for two-dimensional nonlinear dispersion Boussinesq equation Ⅱ. Solitary pattern solutions, Chaos, Solitons and Fractals, 18 (2003), 869-880.  doi: 10.1016/S0960-0779(03)00059-6.  Google Scholar

[23]

Ya. B. Zel'dovich, The motion of a gas under the action of a short term pressure shock, Akust. Zh., 2 (1956), 28–38; Soviet Phys. Acoustics, 2 (1956), 25–35.  Google Scholar

show all references

References:
[1]

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.  Google Scholar

[2]

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, 10 (2013), 1761-1792.  doi: 10.1007/s00009-013-0263-3.  Google Scholar

[3]

P. Álvarez-Caudevilla and V. A. Galaktionov, Well-posedness of the Cauchy problem for a fourth-order thin film equation via regularization approaches, Nonlinear Analysis, 121 (2015), 19-35.  doi: 10.1016/j.na.2014.08.002.  Google Scholar

[4]

K. Deimling, Nonlinear Functional Analysis, Springer-Verlag, Berlin/Tokyo, 1985. doi: 10.1007/978-3-662-00547-7.  Google Scholar

[5]

J. Eggers and M. Fontelos, The role of self-similarity in singularities of partial differential equations, Nonlinearity, 22 (2009), R1-R44.  doi: 10.1088/0951-7715/22/1/R01.  Google Scholar

[6]

Yu. V. Egorov, V. A. Galaktionov, V. A. Kondratiev and S. I. Pohozaev, Asymptotic behaviour of global solutions to higher-order semilinear parabolic equations in the supercritical range, Adv. Differ. Equat., 9 (2004), 1009-1038.   Google Scholar

[7]

J. D. Evans, V. A. Galaktionov and J. R. King, Unstable sixth-order thin film equation. Ⅰ. Blow-up similarity solutions; Ⅱ. Global similarity patterns, Nonlinearity, 20 (2007), 1799-1841, 1843-1881.  doi: 10.1088/0951-7715/20/8/003.  Google Scholar

[8]

A. Favini, G. R. Goldstein, J. A. Goldstein and S. Romanelli, Classification of general Wentzell boundary conditions for fourth order operators in one space dimension, J. Math. Anal. Appl., 333 (2007), 219-235.  doi: 10.1016/j.jmaa.2006.11.058.  Google Scholar

[9]

V. A. Galaktionov, On higher-order viscosity approximations of odd-order nonlinear PDEs, J. Engr. Math., 60 (2008), 173-208.  doi: 10.1007/s10665-007-9146-6.  Google Scholar

[10]

V. A. Galaktionov, Hermitian spectral theory and blow-up patterns for a fourth-order semilinear Boussinesq equation, Stud. Appl. Math., 121 (2008), 395-431.  doi: 10.1111/j.1467-9590.2008.00421.x.  Google Scholar

[11]

V. A. Galaktionov, Formation of shocks in higher-order nonlinear dispersion PDEs: nonuniqueness and nonexistence of entropy, (2009), arXiv: 0902.1635. Google Scholar

[12]

V. A. Galaktionov, Regional, single point, and global blow-up for the fourth-order porous medium type equation with source, J. Partial Differ. Equ., 23 (2010), 105–146, arXiv: 0901.4279.  Google Scholar

[13]

V. A. Galaktionov, Shock waves and compactons for fifth-order nonlinear dispersion equations, European J. Appl. Math., 21 (2010), 1–50. (arXiv: 0902.1632). doi: 10.1017/S0956792509990118.  Google Scholar

[14]

V. A. Galaktionov, Single point gradient blow-up and nonuniqueness for a third-order nonlinear dispersion equation, Stud. Appl. Math., 126 (2011), 103-143.  doi: 10.1111/j.1467-9590.2010.00499.x.  Google Scholar

[15]

V. A. Galaktionov and S. I. Pohozaev, Third-order nonlinear dispersive equations: Shocks, rarefaction, and blow-up waves, Comput. Math. Math. Phys., 48 (2008), 1784-1810.  doi: 10.1134/S0965542508100060.  Google Scholar

[16]

V. A. Galaktionov and S. R. Svirshchevskii, Exact Solutions and Invariant Subspaces of Nonlinear Partial Differential Equations in Mechanics and Physics, Chapman & Hall/CRC, Boca Raton, Florida, 2007.  Google Scholar

[17]

M. Inc, New compacton and solitary pattern solutions of the nonlinear modified dispersive Klein-Gordon equations, Chaos, Solitons and Fractals, 33 (2007), 1275-1284.  doi: 10.1016/j.chaos.2006.01.083.  Google Scholar

[18]

M. A. Krasnosel'skii and P. P. Zabreiko, Geometrical Methods of Nonlinear Analysis, Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69409-7.  Google Scholar

[19]

A. S. Markus, Introduction to the Spectral Theory of Polynomial Operator Pencils, Transl. Math. Mon., Vol. 71, Amer. Math. Soc., Providence, RI, 1988.  Google Scholar

[20]

M. A. Naimark, Linear Differential Operators, Part Ⅰ, Ungar Publ. Comp., New York, 1968.  Google Scholar

[21]

M. A. Vainberg and V. A. Trenogin, Theory of Branching of Solutions of Non-Linear Equations, Noordhoff Int. Publ., Leiden, 1974.  Google Scholar

[22]

Z. Yan, Constructing exact solutions for two-dimensional nonlinear dispersion Boussinesq equation Ⅱ. Solitary pattern solutions, Chaos, Solitons and Fractals, 18 (2003), 869-880.  doi: 10.1016/S0960-0779(03)00059-6.  Google Scholar

[23]

Ya. B. Zel'dovich, The motion of a gas under the action of a short term pressure shock, Akust. Zh., 2 (1956), 28–38; Soviet Phys. Acoustics, 2 (1956), 25–35.  Google Scholar

Figure 1.  Illustrative numerical solutions of the oscillatory function $H(s)$ from (2.8) in the case $n = 1$ and selected $\beta$
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Figure 2.  Shooting the first similarity profile satisfying (5.5), (5.7) for $n = 1$
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Figure 3.  he first similarity profiles satisfying (5.5), (5.7) for $n = 3, 2, 1, 0$ and $n = -0.5$
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">Figure 4.  Numerical solution in the $n=0$ case of (3.1) with (3.7) and $\nu=g"'(0)=-1,\alpha=0.5$
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