# American Institute of Mathematical Sciences

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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:

show all references

##### References:
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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