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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August
2018, 38(8): 3913-3938.
doi: 10.3934/dcds.2018170
Gradient blow-up for a fourth-order quasilinear Boussinesq-type equation
| 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 |
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
| $n \to 0^+$ |
| $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 and 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. |
| [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. |
| [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. |
| [4] |
K. Deimling,
Nonlinear Functional Analysis, Springer-Verlag, Berlin/Tokyo, 1985.
doi: 10.1007/978-3-662-00547-7. |
| [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. |
| [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.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [11] |
V. A. Galaktionov, Formation of shocks in higher-order nonlinear dispersion PDEs: nonuniqueness and nonexistence of entropy,
(2009), arXiv: 0902.1635. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [19] |
A. S. Markus,
Introduction to the Spectral Theory of Polynomial Operator Pencils, Transl. Math. Mon., Vol. 71, Amer. Math. Soc., Providence, RI, 1988. |
| [20] |
M. A. Naimark,
Linear Differential Operators, Part Ⅰ, Ungar Publ. Comp., New York, 1968. |
| [21] |
M. A. Vainberg and V. A. Trenogin,
Theory of Branching of Solutions of Non-Linear Equations, Noordhoff Int. Publ., Leiden, 1974. |
| [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. |
| [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. |
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. |
| [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. |
| [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. |
| [4] |
K. Deimling,
Nonlinear Functional Analysis, Springer-Verlag, Berlin/Tokyo, 1985.
doi: 10.1007/978-3-662-00547-7. |
| [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. |
| [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.
|
| [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. |
| [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. |
| [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. |
| [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. |
| [11] |
V. A. Galaktionov, Formation of shocks in higher-order nonlinear dispersion PDEs: nonuniqueness and nonexistence of entropy,
(2009), arXiv: 0902.1635. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [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. |
| [19] |
A. S. Markus,
Introduction to the Spectral Theory of Polynomial Operator Pencils, Transl. Math. Mon., Vol. 71, Amer. Math. Soc., Providence, RI, 1988. |
| [20] |
M. A. Naimark,
Linear Differential Operators, Part Ⅰ, Ungar Publ. Comp., New York, 1968. |
| [21] |
M. A. Vainberg and V. A. Trenogin,
Theory of Branching of Solutions of Non-Linear Equations, Noordhoff Int. Publ., Leiden, 1974. |
| [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. |
| [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. |
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