we study the instability of a gravitational system composed of thermodynamic gas or star system as collisionless particles with dark matter background. We analyze qualitatively the dispersion relation curves and discuss the probable formes about damping and collapse solution. The analytic formulae of the damping and collapse time can be obtained. The results show that the existence of non-baryonic dark matter background influences seriousely the instability of a gravitational system.
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