Publication: Gradient blow-up for a fourth-order quasilinear Boussinesq-type equation
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2018-08
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American Institute of Mathematical Sciences (AIMS)
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Abstract
The Cauchy problem for a fourth-order Boussinesq-type quasilinear wave equation (QWE-4) of the form u(tt) = -(vertical bar u vertical bar(n) u)(xxxx) in R x R+, 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 -> 0(+), when the classic linear beam equation occurs u(tt) = - u(xxxx), with simple, better-known and understandable evolution properties.
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Fourth-order quasilinear wave equation, Gradient blow-up, Self- similarity of the second kind
Bibliographic citation
Álvarez-Caudevilla, P., D. Evans, J. & A. Galaktionov, V. (2018). Gradient blow-up for a fourth-order quasilinear Boussinesq-type equation. Discrete & Continuous Dynamical Systems, 38(8), pp. 3913–3938.