For quark matter at finite baryon chemical potential, its density develops large fluctuations when it undergoes a first-order phase transition. Based on the NambuJonaLasinio (NJL) model, we have used the linear response theory to study the growth rate of density fluctuations and its dependence on the wavelength of unstable modes. Using the transport equation derived from the NJL model, we have also studied the time evolution of the unstable modes and the density fluctuations in a baryon-rich quark matter that is confined in a finite volume. Allowing the expansion of the quark matter, we have further studied the survivability of the density fluctuations as the density and temperature of the quark matter decrease. Possible experimental signatures of the density fluctuations have been suggested.