The Cauchy problem for a nonlinear Sobolev equation theories of ferroelectricity

This problem arises when considering an anisotropic ferroelectric near the Curie temperature with anisotropy along the Oz axis and taking into account the model time dispersion. The main results of the study: 1. A fundamental solution is constructed and its exact estimates are obtained. 2. The properties of the volume potential are studied. 3. On the basis of potential theory, conditions are obtained under which a weak generalized solution of the Cauchy problem exists. 4. Conditions for the instantaneous blow-up of a weak solution of the Cauchy problem are obtained.