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We use null isometric immersions to introduce time-dependent null hypersurfaces, in a Lorentzian manifold, evolving in the direction of their mean curvature vector (a vector transversal to the null hypersurface). We prove an existence result for such hypersurfaces in a short-time interval. Then, we discuss the evolution of some induced geometric objects. Consequently, we prove under certain geometric conditions that some of the above objects will blow-up in finite time. Also, several examplesdoi:10.4064/cm7450-8-2018 fatcat:pkh7btgtanekdelox76e6dt5ka