Hermitian Spectral Theory and Blow-Up Patterns for a Fourth-Order Semilinear Boussinesq Equation

Reference:

Galaktionov, V. A., 2008. Hermitian Spectral Theory and Blow-Up Patterns for a Fourth-Order Semilinear Boussinesq Equation. Studies in Applied Mathematics, 121 (4), pp. 395-431.

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Official URL:

http://dx.doi.org/10.1111/j.1467-9590.2008.00421.x

Abstract

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.

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