  Rigorous results on the nonlinear dynamics of a dilute Bose gas with a
negative scattering length in an harmonic magnetic trap are presented and
sufficient conditions for the collapse of the system are formulated. By using
the virial theorem for the Gross-Pitaevskii equations in an external field we
analyze the temporal evolution of the mean square radius of the gas cloud. In
the 2D case the quantity undergoes harmonic oscillation with frequency $2\omega
_0$ It implies that for a negative value of energy of the system, the gas cloud
will collapse after a finite time interval. For positive energy the cloud
collapses if the initial conditions correspond to a large enough amplitude of
oscillations. Stable oscillations with a small amplitude are possible. In the
3D case the system also collapsed after a finite time for a state with negative
energy. A stringent condition for the collapse is also derived.
