Abstract: In this paper, we deal with some fast diffusion equations u_t = ?u^m+ au^αv^p and v_t = ?vⁿ+ bu^qv^β subject to null Dirichlet boundary conditions. We prove that every solution vanishes in finite time for pq >（m-α）（n-β）, m > α and n > β, where the exact relation of initial data is determined with the aid of some Sobolev Embedding inequalities.If pq <（m-α）（n-β）, m > α and n > β, we show the barriers of the initial data which lead to the non-extinction of solutions. For the case pq =（m-α）（n-β）, the solutions vanish for small initial data. The results fill in a gap for the case pq < mn in Nonlinear Anal. Real World Appl. 4（2013） 1931-1937. The coefficients a and b play a vital role in the existence of non-extinction weak solution provided that a and b are large enough. At last, we use the scaling methods to determine some exponent regions where one of the components would blow up alone for some suitable initial data. 

Key words: Fast diffusion, Extinction, Non-simultaneous blow-up