Figure 1.
Illustrative numerical solutions of the oscillatory function $H(s)$ from (2.8) in the case $n = 1$ and selected $\beta$
-
Previous Article
Classification of extremal functions to logarithmic Hardy-Littlewood-Sobolev inequality on the upper half space
- DCDS Home
- This Issue
-
Next Article
Oscillating solutions for prescribed mean curvature equations: euclidean and lorentz-minkowski cases
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. |
| [1] |
Jinxing Liu, Xiongrui Wang, Jun Zhou, Huan Zhang. Blow-up phenomena for the sixth-order Boussinesq equation with fourth-order dispersion term and nonlinear source. Discrete and Continuous Dynamical Systems - S, 2021, 14 (12) : 4321-4335. doi: 10.3934/dcdss.2021108 |
| [2] |
Filippo Gazzola, Paschalis Karageorgis. Refined blow-up results for nonlinear fourth order differential equations. Communications on Pure and Applied Analysis, 2015, 14 (2) : 677-693. doi: 10.3934/cpaa.2015.14.677 |
| [3] |
Claudia Anedda, Giovanni Porru. Second order estimates for boundary blow-up solutions of elliptic equations. Conference Publications, 2007, 2007 (Special) : 54-63. doi: 10.3934/proc.2007.2007.54 |
| [4] |
Mohammad Kafini. On the blow-up of the Cauchy problem of higher-order nonlinear viscoelastic wave equation. Discrete and Continuous Dynamical Systems - S, 2022, 15 (5) : 1221-1232. doi: 10.3934/dcdss.2021093 |
| [5] |
Jong-Shenq Guo. Blow-up behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete and Continuous Dynamical Systems, 2007, 18 (1) : 71-84. doi: 10.3934/dcds.2007.18.71 |
| [6] |
Takiko Sasaki. Convergence of a blow-up curve for a semilinear wave equation. Discrete and Continuous Dynamical Systems - S, 2021, 14 (3) : 1133-1143. doi: 10.3934/dcdss.2020388 |
| [7] |
Donghao Li, Hongwei Zhang, Shuo Liu, Qingiyng Hu. Blow-up of solutions to a viscoelastic wave equation with nonlocal damping. Evolution Equations and Control Theory, 2022 doi: 10.3934/eect.2022009 |
| [8] |
Olivier Druet, Emmanuel Hebey and Frederic Robert. A $C^0$-theory for the blow-up of second order elliptic equations of critical Sobolev growth. Electronic Research Announcements, 2003, 9: 19-25. |
| [9] |
Mikhaël Balabane, Mustapha Jazar, Philippe Souplet. Oscillatory blow-up in nonlinear second order ODE's: The critical case. Discrete and Continuous Dynamical Systems, 2003, 9 (3) : 577-584. doi: 10.3934/dcds.2003.9.577 |
| [10] |
Frédéric Abergel, Jean-Michel Rakotoson. Gradient blow-up in Zygmund spaces for the very weak solution of a linear elliptic equation. Discrete and Continuous Dynamical Systems, 2013, 33 (5) : 1809-1818. doi: 10.3934/dcds.2013.33.1809 |
| [11] |
Bouthaina Abdelhedi, Hatem Zaag. Single point blow-up and final profile for a perturbed nonlinear heat equation with a gradient and a non-local term. Discrete and Continuous Dynamical Systems - S, 2021, 14 (8) : 2607-2623. doi: 10.3934/dcdss.2021032 |
| [12] |
Zhiqing Liu, Zhong Bo Fang. Blow-up phenomena for a nonlocal quasilinear parabolic equation with time-dependent coefficients under nonlinear boundary flux. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3619-3635. doi: 10.3934/dcdsb.2016113 |
| [13] |
Evgeny Galakhov, Olga Salieva. Blow-up for nonlinear inequalities with gradient terms and singularities on unbounded sets. Conference Publications, 2015, 2015 (special) : 489-494. doi: 10.3934/proc.2015.0489 |
| [14] |
Marius Ghergu, Vicenţiu Rădulescu. Nonradial blow-up solutions of sublinear elliptic equations with gradient term. Communications on Pure and Applied Analysis, 2004, 3 (3) : 465-474. doi: 10.3934/cpaa.2004.3.465 |
| [15] |
Jibin Li, Yan Zhou. Bifurcations and exact traveling wave solutions for the nonlinear Schrödinger equation with fourth-order dispersion and dual power law nonlinearity. Discrete and Continuous Dynamical Systems - S, 2020, 13 (11) : 3083-3097. doi: 10.3934/dcdss.2020113 |
| [16] |
Alberto Bressan, Massimo Fonte. On the blow-up for a discrete Boltzmann equation in the plane. Discrete and Continuous Dynamical Systems, 2005, 13 (1) : 1-12. doi: 10.3934/dcds.2005.13.1 |
| [17] |
Yuta Wakasugi. Blow-up of solutions to the one-dimensional semilinear wave equation with damping depending on time and space variables. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3831-3846. doi: 10.3934/dcds.2014.34.3831 |
| [18] |
Min Li, Zhaoyang Yin. Blow-up phenomena and travelling wave solutions to the periodic integrable dispersive Hunter-Saxton equation. Discrete and Continuous Dynamical Systems, 2017, 37 (12) : 6471-6485. doi: 10.3934/dcds.2017280 |
| [19] |
Asma Azaiez. Refined regularity for the blow-up set at non characteristic points for the vector-valued semilinear wave equation. Communications on Pure and Applied Analysis, 2019, 18 (5) : 2397-2408. doi: 10.3934/cpaa.2019108 |
| [20] |
Xiaoqiang Dai, Chao Yang, Shaobin Huang, Tao Yu, Yuanran Zhu. Finite time blow-up for a wave equation with dynamic boundary condition at critical and high energy levels in control systems. Electronic Research Archive, 2020, 28 (1) : 91-102. doi: 10.3934/era.2020006 |
2020 Impact Factor: 1.392
Tools
Metrics
Other articles
by authors
[Back to Top]