On Cauchy problems for nonlinear Sobolev equations in ferroelectricity theory

Two Cauchy problems for the nonlinear Sobolev equations $\frac{\partial^2}{\partial t^2}\frac{\partial^2u}{\partial x^2_3}+\Delta u=|u|^q$ and $\frac{\partial^2}{\partial t^2}\Delta_\perp u+\Delta u=|u|^q$ are investigated. Conditions are found under which the Cauchy problems have weak generalized local-in-time solutions, and the blow-up conditions for weak solutions of these problems are determined.