Upper and lower blow-up rate estimates of a semilinear heat equation with a nonlinear boundary condition

In this paper, we consider the blow-up solutions of a semi-linear heat equation with a Neumann boundary condition, where the nonlinear terms, appear in the differential equation and in the boundary conditions, are of exponential types. We study the effect of the nonlinear terms on the last blow-up behavior. Precisely, the upper (lower) blow-up rate estimates are established by using integral equation method and maximum principle techniques. The results show that the presentence of the reaction and boundary terms has important effects on the upper (lower) blow-up rate estimates forms.