Singularities of a Variational Wave Equation

✍ Scribed by Robert T. Glassey; John K. Hunter; Yuxi Zheng

• Book ID: 102973110
• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 782 KB
• Volume: 129
• Category: Article
• ISSN: 0022-0396
• DOI: 10.1006/jdeq.1996.0111

✦ Synopsis

We analyze several aspects of the singular behavior of solutions of a variational nonlinear wave equation which models orientation waves in a massive nematic liquid crystal director field. We prove that smooth solutions develop singularities in finite time. We construct exact travelling wave solutions with cusp singularities, and use them to illustrate a phenomena of accumulation and annihilation of oscillations in sequences of solutions with bounded energy. We also prove that constant solutions of the equation are nonlinearly unstable.

1996 Academic Press, Inc. smooth solutions to the Cauchy problem for (1.1) follows by standard arguments (see [13], for example). The purpose of this paper is to prove that (1.1) does not have global smooth solutions for general smooth initial data, and that the derivatives u t , u x typically become infinite in finite time.