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Hermitian Spectral Theory and Blow‐Up Patterns for a Fourth‐Order Semilinear Boussinesq Equation
Author(s) -
Galaktionov V. A.
Publication year - 2008
Publication title -
studies in applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.164
H-Index - 46
eISSN - 1467-9590
pISSN - 0022-2526
DOI - 10.1111/j.1467-9590.2008.00421.x
Subject(s) - mathematics , eigenfunction , mathematical analysis , hermitian matrix , quadratic equation , hermite polynomials , wave equation , interlacing , nonlinear system , type (biology) , pure mathematics , eigenvalues and eigenvectors , physics , ecology , geometry , quantum mechanics , biology , computer science , operating system
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, equationThe 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 equationwhich was known to admit blow‐up solutions since Keller's work in 1957.