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

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.