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 language | English |
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Pages (from-to) | 3913-3938 |
Number of pages | 26 |
Journal | Discrete and Continuous Dynamical Systems- Series A |
Volume | 38 |
Issue number | 8 |
DOIs | |
Publication status | Published - 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