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: |
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
K. Deimling,
Nonlinear Functional Analysis, Springer-Verlag, Berlin/Tokyo, 1985.
doi: 10.1007/978-3-662-00547-7.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
V. A. Galaktionov, Formation of shocks in higher-order nonlinear dispersion PDEs: nonuniqueness and nonexistence of entropy,
(2009), arXiv: 0902.1635.
![]() |
|
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.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
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.![]() ![]() ![]() |
|
A. S. Markus,
Introduction to the Spectral Theory of Polynomial Operator Pencils, Transl. Math. Mon., Vol. 71, Amer. Math. Soc., Providence, RI, 1988.
![]() ![]() |
|
M. A. Naimark,
Linear Differential Operators, Part Ⅰ, Ungar Publ. Comp., New York, 1968.
![]() ![]() |
|
M. A. Vainberg and V. A. Trenogin,
Theory of Branching of Solutions of Non-Linear Equations, Noordhoff Int. Publ., Leiden, 1974.
![]() ![]() |
|
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.![]() ![]() ![]() |
|
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
Shooting the first similarity profile satisfying (5.5), (5.7) for
he first similarity profiles satisfying (5.5), (5.7) for
Numerical solution in the