Study of the Blow-Up of Solutions of Systems of Equations of Hydrodynamic Type

E. V. Yushkov

M. V. Lomonosov Moscow State University

Abstract: We study the initial boundary-value problem for three-dimensional systems of equations of pseudoparabolic type. The system is similar to the Oskolkov system, but differs from it by the presence of a source of arbitrary order and of a nonlinearity multiplying the highest derivative with respect to time. The local (in time) solvability of the problem is proved in the weak generalized sense. Sufficient conditions for the global (in time) solvability are obtained. We find estimates for the existence time of the solution and for the initial function associated with the blow-up of the solution in finite time.

Keywords: equations of hydrodynamic type, pseudoparabolic equation, initial boundary-value problem, blow-up of a solution, bounded Lipschitz domain, Sylvester's criterion.

UDC: 517.957

Received: 07.04.2010
Revised: 14.09.2010

DOI: 10.4213/mzm8787

 English version: Mathematical Notes, 2011, 90:4, 597–610

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