On the blowup of solutions of the Cauchy problem for nonlinear equations of ferroelectricity theory

We study two Cauchy problems for nonlinear equations of the Sobolev type, of the form $ \frac{\partial}{\partial t}\frac{\partial^2u}{\partial x_3^2} + \Delta u=|u|^q $ and $ \frac{\partial}{\partial t}\Delta_{\perp}u + \Delta u= |u|^q$. We find conditions under which weak generalized local-in-time solutions of the Cauchy problem exist, and we also find conditions under which solutions blow up.