Abstract: This paper deals with the blow-up properties of solutions to semilinear heat equation u_t-u_xx=u^p in (0, 1)×(0, T) with the Neumann boundary condition u_x(0, t)=0, u_x(1, t)=1 on[0, T). The necessary and sufficient conditions under which all solutions to have a finite time blow-up and the exact blow-up rates are established. It is proved that the blow-up will occur only at the boundary x=1. The asymptotic behavior near the blow-up time is also studied.

Key words: semilinear heat equation, Neumann boundary conditions, blow-up rate, blow-up point, blow-up limit