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

The Cauchy problem for a fourth-order Boussinesq-type quasilinear wave equation (QWE–4) of the form utt = −(|u|ⁿu)xxxx in R × 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 utt = −uxxxx, with simple, better-known and understandable evolution properties.