Abstract: This paper gives the sufficient and necessary conditions of existence of global solutions and decay estimates of the solutions for the initial boundary value problem of some nonlinear parabolic equations with small initial energy and the nonlinear power less than Sobolev critical value. The existence, nonexistence and the decay estimates of global solutions are considered. The conditions that initial energy is small and nonlinear power is less than Sobolev critical value is imposed.

Key words: nonlinear parabolic equation, initial boundary value problem, global solution