“Destruction” of the Solutions of Sobolev-Type Nonlinear Wave Equations with Cubic Sources.

We consider model three-dimensional wave nonlinear equations of Sobolev type with cubic sources and, first of all, model three-dimensional equations of Benjamin-Bona-Mahony type and of Rosenau type with model cubic sources. We also study an essentially three-dimensional nonlinear equation of spin waves with cubic source. For these equations, we consider the first initial boundary-value problem in a bounded domain with smooth boundary. We prove local solvability in the strong generalized sense and, for an equation of Benjamin-Bona-Mahony type with source, we prove the unique solvability of the “weakened” solution. We obtain sufficient conditions for the “destruction” of the solutions to the problems under consideration. These conditions mean that the value of the initial perturbation in the norms of some Banach spaces is “large.” Finally, for an equation of Benjamin-Bona-Mahony type, we prove that the “weakened” solution fails in a finite time. [ABSTRACT FROM AUTHOR]