The boundary quenching behavior of a semilinear parabolic equation

In this paper we consider the boundary quenching behavior of a semilinear parabolic problem in one-dimensional space, of which the nonlinearity appears both in the source term and in the Neumann boundary condition. First we proved that the solution quenches at somewhere in some finite time. Then we assert that the quenching can only occur on the left boundary if the given initial datum is monotone. Finally we derived the upper and lower bounds for the quenching rate of the solution near the quenching time. Thus we generalized our former results.