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

Pablo Álvarez-Caudevilla, Jonathan D. Evans, Victor A. Galaktionov

Research output: Contribution to journalArticle

Abstract

The Cauchy problem for a fourth-order Boussinesq-type quasilinear wave equation (QWE–4) of the form utt = −(|u|nu)xxxx in R × 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 utt = −uxxxx, with simple, better-known and understandable evolution properties.

Original languageEnglish
Pages (from-to)3913-3938
Number of pages26
JournalDiscrete and Continuous Dynamical Systems- Series A
Volume38
Issue number8
DOIs
Publication statusPublished - 1 Aug 2018

Keywords

  • Fourth-order quasilinear wave equation
  • Gradient blow-up
  • Self-similarity of the second kind

ASJC Scopus subject areas

  • Analysis
  • Discrete Mathematics and Combinatorics
  • Applied Mathematics

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