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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

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.
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 - A, 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.
Illustrative numerical solutions of the oscillatory function $H(s)$ from (2.8) in the case $n = 1$ and selected $\beta$
Shooting the first similarity profile satisfying (5.5), (5.7) for $n = 1$
he first similarity profiles satisfying (5.5), (5.7) for $n = 3, 2, 1, 0$ and $n = -0.5$
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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