This paper deals mainly with the problem of existence and decay of the global so-lution of a nonlinear degenerated parabolic equation system. The main results are: if α_ij<p_i-1(i,j=1.m) and u_i⁰(x)∈L^p₀-2(Ω),then, (E1)admits a global solution u(x,t),and if α_ij>p_i-1(i,j=1.m) and u_i⁰(x)∈L^p₀-2(Ω),then,there exists d>0 such that ||u_i⁰||_p0-2<d₀(i=1.m), the problem(E1)admits a global solution u(x,t).