Carleman estimate and null controllability for a degenerate parabolic equation with a slightly superlinear reaction term

This paper concerns the null controllability of an one-dimensional degenerate parabolic equation with a slightly superlinear reaction term in a bounded interval, with a control on a subinterval. The degeneracy of the equation is relatively weak, and the reaction term grows relatively slow at infinity. It is noted that the behavior of the reaction term at infinity is consistent with the well-known null controllability results for heat equations. The Carleman estimate and the observability inequality for the corresponding linear conjugate equations are established. Subsequently, the control problem is proved to be null controllable by using the observability inequality, the the Kakutani fixed point theorem and some a priori estimates.