Solution blowup for nonlinear equations of the Khokhlov–Zabolotskaya type

We consider several nonlinear evolution equations sharing a nonlinearity of the form $\partial^2\!u^2/\partial t^2$. 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.