Gradient blow-up for a fourth-order quasilinear Boussinesq-type equation

Pablo Álvarez-Caudevilla; Jonathan D. Evans; Victor A. Galaktionov

doi:10.3934/dcds.2018170

Article Contents

2018, Volume 38, Issue 8: 3913-3938. Doi: 10.3934/dcds.2018170

This issue Previous Article Oscillating solutions for prescribed mean curvature equations: euclidean and lorentz-minkowski cases Next Article Classification of extremal functions to logarithmic Hardy-Littlewood-Sobolev inequality on the upper half space

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: August 31, 2017

Revised: January 31, 2018

Published: May 2018

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

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Abstract

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.

Keywords:

Mathematics Subject Classification: 41A60, 35C20, 35G20, 35C06.

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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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Figure 2. Shooting the first similarity profile satisfying (5.5), (5.7) for $n = 1$

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Figure 3. he first similarity profiles satisfying (5.5), (5.7) for $n = 3, 2, 1, 0$ and $n = -0.5$

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Figure 4. 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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