Abstract: This paper concerns with the initial and boundary value problem for a nonlinear diffusion equation with degeneracies. Based on the proof of existence of weak solutions and vanishing viscosity method, the concept of solutions of regularization is introduced, the meirt of which is the uniqueness of solutions of regularization is guaranteed, moreover, under some conditions, the existence is also established. Subsequently, as a important property, the time invariance of support set of solutions of regularization is proved. Also an explicit solution of regularization to a special problem is constructed. Finally, by the method of energy estimation, a long time behavior of the solution of regularization is studied.

Key words: degeneracy, nonlinear diffusion equation, solution of regularization, uniqueness, localization, long time behavior