Solution Blowup for Nonlinear Equations of the Khokhlov–Zabolotskaya Type
We consider several nonlinear evolution equations sharing a nonlinearity of the form ∂2u2/∂t2. Such a nonlinearity is present in the Khokhlov–Zabolotskaya equation, in other equations in the theory of nonlinear waves in a fluid, and also in equations in the theory of electromagnetic waves and ion–sound waves in a plasma. We consider sufficient conditions for a blowup regime to arise and find initial functions for which a solution understood in the classical sense is totally absent, even locally in time, i.e., we study the problem of an instantaneous blowup of classical solutions.
Keywordsfinite-time blowup nonlinear wave instantaneous blowup
Unable to display preview. Download preview PDF.
- 1.S. N. Gurbatov, O. V. Rudenko, and A. I. Saichev, Waves and Structures in Nonlinear Nondispersive Media: Applications to Nonlinear Acoustics [in Russian], Fizmatlit, Moscow (2008); English transl.: Waves and Structures in Nonlinear Nondispersive Media: General Theory and Applications to Nonlinear Acoustics, Springer, Berlin (2011).zbMATHGoogle Scholar
- 3.N. S. Bahvalov, Ya. M. Zhileikin, and E. A. Zabolotskaya, Nonlinear Theory of Sound Beams [in Russian], Nauka, Moscow (1982).Google Scholar