On the ``Destruction'' of Solutions of Nonlinear Wave Equations of Sobolev Type with Cubic Sources

M. O. Korpusov ; A. G. Sveshnikov

Matematičeskie zametki, Tome 78 (2005) no. 4, pp. 559-578.

Résumé

We consider model three-dimensional wave nonlinear equations of Sobolev type with cubic sources, and foremost, model three-dimensional equations of Benjamin–Bona–Mahony and Rosenau types with model cubic sources. An essentially three-dimensional nonlinear equation of spin waves with cubic source is also studied. For these equations, we investigate 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 a “weakened” solution. We obtain sufficient conditions for the “destruction” of the solutions of the problems under consideration. These conditions have the sense of a “large” value of the initial perturbation in the norms of certain Banach spaces. Finally, for an equation of Benjamin–Bona–Mahony type, we prove the “failure” of a “weakened” solution in finite time.

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