Two families of asymptotic blow-up patterns of nonsimilarity and similarity kinds are studied in the Cauchy problem for the fourth-order semilinear wave, or Boussinesq-type, equation The first countable family is constructed by matching with linearized patterns obtained via eigenfunctions (generalized Hermite polynomials) of a related quadratic pencil of linear operators. The second family comprises nonlinear blow-up patterns given by self-similar solutions. The results have their counterparts in the classic second-order semilinear wave equation which was known to admit blow-up solutions since Keller's work in 1957.
Galaktionov, V. A. (2008). Hermitian Spectral Theory and Blow-Up Patterns for a Fourth-Order Semilinear Boussinesq Equation. Studies in Applied Mathematics, 121(4), 395-431. https://doi.org/10.1111/j.1467-9590.2008.00421.x