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

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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  • 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.

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


    \begin{equation} \\ \end{equation}
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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 $\beta$

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

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

    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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