{'fraction_non_alphanumeric': 0.023578363384188627, 'fraction_numerical': 0.005547850208044383, 'mean_word_length': 6.148514851485149, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
Hermitian Spectral Theory and Blow‐Up Patterns for a Fourth‐Order Semilinear Boussinesq Equation

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.