We prove the global existence and gradient estimate of the solution to the initial boundary problem for the quasi-linear parabolic equationut-div｛σ（|u（t）|2）u（t）｝+g（t,x,u,u）=0 in Ω×[0,∞).with the initial-boundary conditions u（x,0）=u0∈W1,p00（Ω）,and u（x,t）Ω=0 for 0≤t<+∞,where Ω is a bounded domain in RN,σ（|u（t）|2）is a function like σ（|u（t）|2）=|u|m,m>0,and（g（t,x,u,u））is a nonlinear perturbation like ts|x|μuα|u|β+1.Under the assumptions that the mean curvature of Ω is nonnegative and ‖u0‖p0,p0≥ max｛2N（β-m）,2αN,2｝is small.We derive in particular an precise estimate for ‖u‖∞,which includes global existence of solution.