Solution Blowup for Nonlinear Equations of the Khokhlov–Zabolotskaya Type

Abstract

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.